What is a good alternative to the family doctor in NYC for STD screening?  

It's free at the Department of Health.  Click the link to find out when and where.  You can test for HIV anonymously, if you prefer.  The drawback is, you can't make an appointment and show up at that time.  You have to show up early, fill out the form, and wait.  A good thing for women, they do a complete pelvic exam, with a Pap smear.

http://www.ci.nyc.ny.us/html/doh/html/std/stdfree.html

Also, there are private clinics that advertise on the back of the Village Voice.