Are you talking about neighborhoods or specific hotels/hostels?
The city itself is very safe; it's the 4th safest big city in the US. I've lived here for 3 years and have yet to have even an inkling of a problem. My favorite neighborhood (and great for a young visitor) is the East Village. Greenwich Village, The Bowery, SoHo, and TriBeCa are all cool neighborhoods too. Let me know if you need specific recommendations! about 10 years ago
I was in NY in June and I stayed at Jazz on the Park. It's a hostel, pretty cheap one next to central park on the upper west side. For the price we payed it was actually pretty good. And the people there were very nice. And we had no problems, security wise. It is not that central, if you're looking for a downtown place but it is pretty accessible, you have the subway just around the corner. about 10 years ago
Yep jazz on the park gets good reviews on hostelworld.com would recommend it too, very central location
http://www.hostelworld.com/hosteldetails.php/Jazz-on-the-Park-Hostel/New-York/2294 about 10 years ago