Most of the hotels in Manhattan have a swimming pool which you can enjoy for free throughout your visit. There are several pool and gymnasium complexes in New York including 342 E 54th Street, 59th Street/West End Avenue and 35 W 134th Street.
Swimming and surfing is extremely popular at the Surfside 3 Maritime Center at Chelsea Piers.
Jones Beach State Park on the Long Island Shoreline is the place to go for a day trip.

Gym/Swimming Pool Complexes:

342 E 54th Street
Tel: 212 397-3147

59th Street/West End Avenue
Tel: 212 397-3120

35 W 134th Street
Tel: 212 397-3193