Advanced parental age, i.e. having an older father, is a risk factor for a wide variety of genetic diseases (e.g. schizophrenia and autism). Advanced maternal age is generally not. In part, this is because older women lose fertility. In part, this is because a woman's eggs are created when she is young, even if they are not available for fertilization until much later, while a man's sperm are created on a regular basis, and so bear the environmental and aging impacts that his body experiences, a mutation risk load that mounts as he gets older.
While it isn't absolutely conclusively, the evidence very strongly points to mutations in sperm cells as the predominant cause of advanced paternal age related conditions, which makes a great deal of sense, simply because there aren't many other mechanisms by which advanced paternal age could cause diseases with a strong hereditary component. The only additional possible source is closely related - epigenetic impacts acquired by a father during life or at birth that are passed on to a child.
Having an older father is not an entirely bad thing, however, from an evolutionary genetics perspective. An older father, by definition, was fit enough to have live to be reasonably old, and to be fertile enough and able to successfully secure a mother for his child at that age. Evolutionarily speaking, older fathers should, by definition, generally be more fit than the average father, and certainly should be more fit than the average man who is born without regard to whether or not he has children.
There is anecdotal ethnographic evidence that older fathers also tend to have greater success in securing more fit than average mothers for their children. Men who married well the first time may have prospered in part because they had better than average wives. Widowers have often been considered prize catches, at least by parents arranging marriages, and most marriages were arranged from at least the Neolithic era until the last couple of centuries. If children of older fathers also tend to have more fit mothers on average (presumably because their fathers are more fit themselves), then the children receive a double fitness benefit in their genetic inheritance, and the maternal half of that benefit does not carry any elevated mutational risks. Certainly, anyway, there is little evidence that the children of older fathers tend to have less fit than average mothers.
Thus, while children of fathers of advanced paternal age are at greater than average risk of novel, harmful mutations (harmful here meaning evolutionary selective fitness disadvantage), their paternal genetic inheritance will generally be one associated with above average fitness. And, since many novel, harmful mutations lead to miscarriages or congenital defects, children of fathers of advanced paternal age who survive past infancy are not bearing the full cost of the additional mutational risk associated with their father's advanced age. In a more harsh age, marginally fit young children might also have received less than full parental support as resources were devoted instead to more fit siblings - infanticide was common place for much of human history, and only in modern times has it become the norm for all of one's children to survive to adulthood.
Even those children of fathers of advanced parental age who do end up having novel and harmful (but non-deadly) mutations will take a general background of above average fitness in their genetic inheritance together with the novel, harmful mutations. Often, the good will be better than the bad, and as a generally rule there will be both fitness enhancing and fitness reducing genetic traits in the child who also had novel, harmful mutations. Novel fitness enhancing mutations are also probably more likely in children of fathers of advanced paternal age, although the literature seems to suggest that fitness reducing mutations are far more common than fitness enhancing mutations. (Many, if not most, mutations neither enhance nor harm evolutionary fitness.)
Of course, the other disadvantage faced by children of fathers of advanced paternal age is an increased likelihood of being without a father or paternal grandparents when one is relatively young, and if fathers and paternal grandparents enhance one's fitness, in the evolutionary selection sense, then that is another strike against them. There is a school of human evolutionary thought that attributes great selective advantage to the extension of life expectancy long enough that the society has a reasonable number of grandparents alive at any given time, based on circumstantial evidence related to life expectancy and population sizes, but it isn't obvious that the benefits of having elders in a community are primarily a benefit to their own grandchildren, or are a benefit to the community generally, for example, by providing it with continuity of knowledge and culture.
Now, the babies of the brood of an older father are still at a disadvantage relative to their older siblings who may have all of their father's genetic fitness, because they will have none of the novel, harmful mutations associated with a father of advanced paternal age, and because they will have received the benefits of having a living father, and perhaps living grandparents, for a larger share of their life. The only real advantage that the babies of the brood have are the possibility that their mother is a subsequent more fit spouse than their older siblings, and the possibility that the parents improved their parenting skills over time after making mistakes the first times around. The babies may also compensate for possibly less paternal parenting with more older sibling support.
The source of fitness for an older father may be different now than it was in pre-modern times.
In pre-modern times, the main genetic windfall one might receive from an older father might have been the good health related traits that allowed him to live until forty years of age or older, while a significant share of his peers died by then, or become infertile by then, or ceased to be able to attract any women to be mothers for their children at that point.
In modern times, in contrast, postponing fatherhood is usually closely associated with having obtained an advanced education before marrying, and having developed a career and obtaining socio-economic success. Simply living to be that old and being able to be fertile enough with reproductive medicine to have children at an advanced age is now common place and has only trivial fitness implications, but attracting a fertile woman to be a mother for your children, and having a reason to defer childbearing is a pursuit that strongly favors traits that favor socio-economic success. Also, the likelihood of an older father dying before a child comes of age is now greatly reduced from pre-modern times, and marriages entered into when a father is more mature and economically successful tend to be more stable (although there are certainly a significant number of serial divocees who have children in each new marriage among older fathers). Older fathers are likely to have more children overall as well, possible forcing each child to receive a smaller share of inherited wealth, although it isn't obvious that property inheritance at death is a very important factor in life success these days.
Another significant subset of men who are more likely than average to be older fathers are men who seek to have as many children as possible for religious reasons. Razib at Gene Expression has made some notable posts exploring the demographic impacts of this trend in affluent societies. Basically, in poor societies, almost everyone is religious and almost everyone has as many children as they can in the hope that some will survive. With economic development, the vast majority of people become less religious and reduce their childbearing rates to replacement levels in short order in order to maximize the economic resources that they can devote to each child because they can have faith that the child will live to be an adult and have children of his or her own. But, a minority of people in developed societies make a conscious, usually religiously motivated choice to have as many children as they can anyway, and that choice has a major impact of the demographics of developed societies (which universally are not willing to let children in big families start and die due to a shortfall of resources, although China did take a very punitive stance to discourage the large family option as it developed with its one child policy for urban families). These demographic trends also have population genetic implications. From an evolutionary fitness perspective having as many kids as you possibly can even in an economically developed modern society is a great choice for you and your co-religionists (although it may be bad for society as a whole), so from an evolutionary selection perspective, an older father who religiously believes in having many children and passed that cultural trait on to his children has caused them to be much more evolutionarily fit.
Also, of course, most children of older fathers don't have any novel harmful mutations. They get their parents above average fitness while bearing not of the hardships associated with having a father of advanced paternal age (except that a sibling may have such issues and that may stress family resources, but that is rarely a deal breaker in modern societies, and was often resolved by the premature death of the unfit sibling in pre-modern societies). This means that the new child enhances the father's evolutionary fitness, on average, making the ex ante decision to have more children an easy one from a selection perspective, and leave the child fit as well.
All of this is very general, but it may also help to provide a general explanation of why genetic mutations associated with advanced paternal age (even in subsequent generations for children whose parents are young), may tend to receive counterbalancing selective advantages that have no direct causal relationship to any novel, harmful mutations that they may have received because they are descendant from a father of advanced parental age.
Indeed, some of the upsides associated with conditions associated with advanced paternal age such as creativity or logical aptitude or enhanced courtship traits, may be products of having more fit than average patriline ancestors rather than of something flowing from the advanced paternal age condition itself. If so, one would expect the upside associated with advanced paternal age conditions to present more strongly in first generation people with those conditions than in subsequent generations where a fit paternal genetic inheritance that came with the novel harmful mutations has been diluted.
As a footnote, with the genetic risk associated with having inbred children is largely avoidable by someone interested in having children, simply by choosing an unrelated mate, which is something that if often possible in modern societies, once you are an older father, you can't undo the decisions to have or not have children that you made earlier in life. Those are sunk costs and the only decision that a prospective older father has to make is to have or not have children. There is no easy mutation risk reducing option at that point. This also makes it an easier choice and helps explain why of two parenting choices that present similar genetic condition risk to offspring, one is usually prohibited legally and discouraged socially, while the other is rarely prohibited legally.