Censorship, from whatever origin, is more common in falling-crime times because it is a more idea-focused culture, so that danger comes from ideas, beliefs, even words or phrases. A rising violence level reminds people of their corporeality, and turns their mind more toward dangerous substances and acts, for example a war on drugs, alcohol, etc.
I've been meaning to write that post for awhile, but for now, take just one case study of greater censorship in falling-crime times -- book-burning. Wikipedia has compiled a list, although for our purposes you should skip down at least to #29 to get into the late Middle Ages. I'm ignoring cases where one civilization burns another's texts, as that could be part of a larger program of inter-group cultural destruction, not censorship within a society. And since we only have historical homicide data for Europe, I'm ignoring China, India, etc.
Notice where there are decade-long gaps.
Somewhere between 1450 and 1550 the European homicide rate began a secular decline that continues through today. However, there have been several reversals of that decline: from ca. 1580 to 1630 (the Elizabethan-Jacobean period), 1780 to 1830 (the Romantic-Gothic period), in some places from 1900 to 1930 (the Jazz Age), and finally from 1960 to 1990 (the New Wave Age).
It's interesting to note that the last hurrah of rising-crime before the secular decline sets in is roughly the 14th C. -- perhaps no surprise if you're familiar with how disastrous in general that period was. Yet there are no book-burning incidents listed, although there are from the 13th and 15th centuries. The Renaissance and Reformation, by which time the secular decline in violence begins, were more in favor of burning books.
There's nothing listed for most of the Elizabethan-Jacobean wave of violence until 1624, either right at the transition to the falling-crime era or perhaps outside of it. The end-points for these reversals are always "circa" of course, because we don't have annual data for homicide rates.
The censors take up their torches again during the mid-17th C. through the mid-18th C. -- the Age of Reason and Enlightenment. There's only one counter-example from the following Romantic-Gothic period, in 1787.
During the Victorian era, it was France that kept the tradition going, along with Gilded Age enthusiasm in America from the New York Society for the Suppression of Vice. During the Jazz Age there are several examples, perhaps reflecting the fact that homicide rates didn't rise across all nations. But in America, where we did have a steady crime wave, there are no incidents listed.
What everyone thinks of by now as the height of book-burning, Nazi Germany, was part of a broader trend during the falling-crime mid-century. It wasn't all political either: they burned comic books across late 1940s America, as part of the moral panic over horror comics. After the last major push, the burning of Wilhelm Reich's works in the mid-late 1950s, there are no incidents from the rising-crime period up through the early '90s.
There's a scene in Footloose where some local busybodies from the church burn corrupting books, when the Reverend Shaw breaks in to tell them to knock it off and go on to look after one another, because Satan is in your heart, not in some book. In the real world, it was Reverend Shaw's view that carried the day during the religious revival of the '60s through the '80s.
In 1988, angry Muslims did burn The Satanic Verses in two English cities, although with so little influence over society at that point, I'm reluctant to treat that as a genuine counter-example, as though a mainstream group had done so. It doesn't look like there were notable burnings during the '90s, although during the 2000s and 2010s they've burned Harry Potter books and Qurans here.
Over the centuries, the scale of each falling-crime wave of book-burning seems to have diminished. That's comforting. Still, perhaps during the past 20 years there would've been a level closer to that of the mid-century if only print and books were as dominant a form of the literate culture as they were back then. With people getting a lot more text from the internet, the censors might not see as much of a point in burning books. But even the Victorian era was a lot better than the Age of Reason / Enlightenment period, let alone the Renaissance-Reformation period.
Probably the censorship in more recent falling-crime periods has just shifted toward regulations and what we call now political correctness, and away from cruder methods like burning books. Sometime later I'll cover the history of censorship codes for popular entertainment; that's one obvious case of substitution of subtler for grosser means of censorship in recent falling-crime eras.
In any case, the main thing to take away is that censorship, judged by the proxy of book burning, rises in a falling-crime period, when people put more faith in the power of ideas and thoughts. When during a rising-crime era they are reminded of the power of weapons, artifacts, substances like drugs, and the body itself, a certain style of moral regulators stops resorting to censorship and more earnestly takes up causes like Prohibition (and during the '80s, raising the drinking age) and the war on drugs, another topic for another post.