What about a bed and breakfast?
http://www.bedandbreakfast.com/new-york/chelsea-lodge.html
http://www.bedandbreakfast.com/new-york/bronx-bed-and-breakfast.html
http://www.bedandbreakfast.com/new-york/connie-billingslea.html
http://www.bedandbreakfast.com/new-york/tony-s-place-bnb.html
http://www.bedandbreakfast.com/new-york/efuru-guest-house.html
http://www.bedandbreakfast.com/manhattan-new-york.html
Can't vouch for them personally but that's the kind of accommodation I used when I stayed in the city.
http://www.bedandbreakfast.com/new-york-city-new-york.html
Blondie