last time I went, I stayed in Long Island City, Queens. Had 4 full days and 2 half days (I went with my sixth form in 2006, but were only there 4 days, so wanted to go back) The subway station (Queensboro Plaza) was one block away, and only about 5 stops into Times Square on the N,W and 7 lines. 7 day ticket is only $25 or something.
Really liked it. It was miles cheaper than staying in Manhattan, breakfast was included and also let you see a little bit more of NYC other than just Manhattan.
Taxi from JFK was about $30 each way, which was nothing when split between 4. Got it off Expedia. Try there.