1. Is it accurate to describe New York City as the safest big city in America?It seems this would be correct. You are still more likely to be a victim of a violent crime in New York than in San Jose, as violent crime rates are higher in NYC.
2. If so, what factors have led to this? In particular, is New York City doing something right that other big cities are doing wrong?The crime trend reversed under Rudy Guiliani, a former prosecutor. But let's not forget that crimes have been greatly reduced everywhere in the US, not just in NYC. NYC has simply seen greater decreases than other cities.
A cynical commentary on the overall decrease in violent crime over the past 15 years is that the turf wars associated with drugs, especially crack cocaine, have largely ended and turfs have been established.
In the mid 1990s, the NYPD began cracking down on all crimes, no matter how small, such as panhandling, vandalism, or jumping subway turnstiles. They also (like many cities in America) put a lot more police on the streets.
An interview with big city police chiefs, 1995. Interesting stuff in there.
It is difficult to say what, if anything, precisely led to the decrease in crime rates. Some say that Roe v. Wade made it easier for mothers who didn't want kids to not have them. Unwanted children are very likely to grow up to be criminals. Some say it was the stabilization of the illegal drug market. Some say the economic boom over the last decade. Some say its because we are locking up prisoners (especially young men) at a much higher rate.
Of course, it may be (and probably is) a combination of all of these things. It really is hard to say. I do think cracking down on minor crimes (possible because of more police on the street) and putting criminals away for longer have the greatest effect.