I am afraid their are not that many, especially for anything bigger than a studio.
Those that you may find are often not legal apartments, or cons, so be carful who you give your money to.
On the search engines just log in for your search Brooklyn, NY and for rent up to $1,000. Or Queens and again up to $1,000 or Bronx etc
Then only look at or call the apartments that have been listed less than a week and close to subways, if you need the subway. Once you have the address; look it up on http://www.NYC.Gov/HPD to see who is listed as the managing agent, be sure to speak to someone in their office or their licensed Broker. All Brokers have a license that looks similar to a drivers license with a photo.
Most of the better apartments in the price range will be a distance from Manhattan, but can be very nice apartments in great neighborhoods.
One of my workers just found a lovely large one bedroom near the Euclid Ave A/C express stop. She thought she would live there to save money to buy a co-op in Fort Greene, now she is thinking she will buy a house out there!