We went for a week last Christmas, and used Airbnb, which worked out brilliantly. Basically, people rent out their houses/apartments, and you usually pay a lot less than a hotel, and have all the flexibility of staying in a proper apartment rather than a hotel.
I don't think the authorities are *too* keen on it, but it's worth checking out. We stayed Downtown not far from Wall Street, and it was very safe. It doesn't take long to get to most places in NYC on the subway.