As an ex-manhattan resident, and the Guy Who Knows New York and Can Tell You Where To Go...
0. Good walking shoes.
1. Bottle of water or two
2. A cheap digital camera + spare batteries
3. A flashlight & Whistle (you should always have that anyway)
4. A pocket knife - any under 4"
5. Pepper Spray
6. A red bandanna
7. A small radio

As far as crime - relax. NYC has more cops on the street than anywere in the USA. It's a full-bore police state, like Singapore. As a result,
NY is safer than Indianapolis Indiana, Phoenix Arizona and just about any other million + sized city. Of course, your civil liberties are limited to being able to put on offensive plays and publishing offensive magazines, but they seem to like that OK in NYC these days.

Don't forget to stop in at Arturo's for good italian food - Houston Street @ Thompson, right there on the corner. Also consider The Noho Star - on Bleeker Street - get the ginger ale, it's a treat.

Also, I'd suggest avoiding the over-priced and underwhelming South Street Seaport, instead, go to the rather entertaining and interesting Chinatown.

The tree is still up - 51st and 6th Ave and the store windows are still all done up. A walk on 57th street is always a trip.

Enjoy, it's a pretty great place, I loved living there when it made sense for me.