Another very good place, although only in the midtown area:

hotels

Their bathrooms are much nicer and cleaner that what you'll find elsewhere - particularly in the subway, where, as a friend put it, "There are bathrooms that Dante himself would not enter."

They are also good places to rest and cool off (or warm up). The main thing is to act as if you belong there, and no one will ask whether you are actually a guest in the hotel - unless you look like a real bum. I have gone into the Plaza Hotel lobby, dressed in cutoffs and a t-shirt, and sat there for a half hour, reading a magazine.