The safety is really not much of a problem, as long as you're not stupid about things. It's in a relatively safe area. As for the school itself, it's overpriced and truly does lack the community. A lot of people have trouble making friends. It's a huge school and that hinders professors from getting personal. The buildings are apparently quite lacking, I've only been in the dorms but they were a bit run down and quite overpriced. Definitely apply, but make sure you visit. A lot of people apply and attend because they are so in love with the city, but quickly become disenchanted.
Something to note - NYU has the highest suicide rate of any college in the US.