I'm a museum junkie. A sucker for the Impressionists. New York is the perfect place for people like me. And if you're smart about it, you can get in free to many museums. New York has many, many other exciting museums, covering nearly every aspect of the human experience.
Several museums offer free admission one night a week, generally during the week.
Most museums offer reduced admission to students and older people. If you live in the metropolitan area and there's a particular museum you visit with any regularity, check to see whether a membership with the museum includes free admission. I got an introductory membership to the Museum of Natural History a few months ago, and after a couple of visits, the membership's already nearly paid for itself. Some museums have a flat fee that you must pay to enter. That's close enough to free for me.