Most of the arguments in favor of a minimum wage are total economic crap. They argue that it is simply immoral to not legislate a minimum wage without reference to its effects. Intentions are more important than results to these people.
However, at least one argument is based on good economics: the argument that labor is insensitive to price. Most often, the law of supply and demand says that when the price goes up, the demand will go down. There are some goods for which this is not true: diamonds or Rolex watches (the real ones; not the ones that spammers want you to buy). At the other end of the curve, it is possible (but for which there is no good empirical evidence) that some tasks simply require a minimally-trained human. Employers will get the same employees as long as the wages remain within this magical range. Thus, minimum wage advocates argue, simple justice requires a minimum wage to ensure that the surplus goes to the workers and not the owners.
The problem is that this theoretical price range probably doesn't exist, or else is so small that it cannot reliably be discerned. Take the task of cleaning a floor. One might use a little four inch diameter broom such as I've seen used in India. Or you might buy a 24 inch push broom. Or a 72 inch floor sweeper. Or a ride-upon vacuum cleaner which sucks the living shit out of anything that might resemble dirt. There are wage scales at which each capital investment makes sense .... or doesn't.
Even if certain market sectors have a price insensitive range, it's quite possible that others do not. A minimum wage law that applies equally to everyone would be a benefit to some workers, and cause workers in other market sectors to have their jobs disappear permanently.
Economics cannot tell you what is moral or not. Economics can only tell you what will happen when you take a given action. Thus, if voters choose to destroy some people's jobs because that is the moral thing to do, they cannot be judged by economists. They can only be judged by their moral authority, who will doubtless send them to their religion's version of hell. But oops, I'm judging them, aren't I? I cannot see how workers are helped by having the worst-off lose their jobs, so that other more prosperous people may earn more money. If a rich person did that, they would be judged harshly by the very people who advocate for a minimum wage.
Oh, and Don Boudreaux points out that a law setting a minimum price for the sale of a car would be rejected by any sane person, and then goes on to ask why said sane people then argue for a minimum price for the sale of labor.