You did not specify that you wanted the name of the cheapest sailing school in NYC but here it is: SailNY - New York City's Community Sailing Association
That said, there are several places you can locate by using a google search of: NYC sailing schools.
I took my basic keelboat course with: New York Sailing Center & Yacht Club: sailing school, club and marina.
on City Island and enjoyed their Benetteau 21' sailboats while sailing around the not too crazy waters of the western LI Sound. Most of the commercial schools are probably done with their last sessions and the first link I posted does some 'frost biting' sailing (cold weather) so they may be your best bet to getting started sooner.
Welcome to SN and the, er, club (asylum, I am pretty sure that SmackDaddy is the, ahem, head nurse - not to be mistaken with the 'Head Mistress', aka, Peggy Hall, located right down the hall: SailboatOwners.com - View Profile: Peggie Hall-HeadMistress
while the principal seems to be on sabbatical, and no one knows for sure just who he or she is.)
If you can guess the movie I am parodying with my parenthetic remarks above you will get points for that and no, there will be no quiz or second guesses.
Welcome to the unstable world of boating and all the bizarre personalities we project on line.
Mostly, have at and have fun.