Hahahhahaha... I can smell the empty subway card phenomena now... Don't miss this, but man, I can smell and identify a creepy homeless guy a block away now.
Seriously though, the neighborhoods people will be mostly in are safe. I still feel safer walking in NYC at night than I do in freaking downtown St. Paul, MN. Fact is, I was actually safer as well.
A couple years ago park slope had a serial rapist. Within a day of the 2nd reported attempt, the cops were all over the place patrolling with lights on at night. It was a little strange, and a lot police state-like, but definitely safe. They did this for a week until they caught the guy. That's the most crime/police issues I remember in 8 years in NYC.
Remember, there are approximately 34,500 police officers in NYC... There are definitely neighborhoods to avoid at night, but I never ended up on one of those, even by accident.