Harlem, Wash. Heights, Inwood, Upper East Side, East Village, Lower East Side, Murray Hill, SoHo, Chinatown, Little Italy, NoLita, Chelsea, Hell's Kitchen. These neighborhoods are some of the best in NYC, and the averaged total of all of these neighborhoods is around 1750$. Enjoy New York!
It of course depends on what you are looking for. There are plenty of somewhat decent apartments in old walk-ups for that amount of money in certain parts of the upper east side or even the upper west side.