With the opening of the new Whitney Museum some people have mentioned to me they don't feel welcome. They find art museums as pretentious and expensive. Art museums exist for the public so no one should feel this way. Many museums regularly have "pay as you wish" and most have regular weekly free nights in order to make everyone in the community feel included.
Do not ever feel like you don't know enough about art to visit a museum. when visiting any museum enjoy the work while respecting it. Always ask before taking photographs and NEVER use a flash or physically touch the work. The oil from our hands as well as flash photography weakens the archival quality of the work. Typically if a work is in the museums permanent collection you may take a non-flash photo. If there is not a docent or tour guide on duty, the museum guards often are incredibly knowledgeable and passionate about the art .
Here are my favorite NYC museums;
WEEKLY FREE DAYS or HOURS:
New Museum of Contemporary Art 7-9 pm
Japan Society 6-9 pm
Brooklyn Botanic Garden 10 am-noon