There used to be many, but the City bought most of them outright and converted them into homeless shelters. The remainder serve as overflow, so you probably don't want to stay there either. The common wisdom in the frequent flyer community seems to be to just head to Manhattan, or go to Flushing - it's more fun that way anyways.
* The exception is probably the newly opened TWA hotel on JFK, which is really nice, albeit pricy