These museums may suggest a nominal fee of, let’s say, $25, however, you can still get full access by handing over as little as a penny.  Tried and true, by Used York City.

American Museum of Natural History
Suggested adult admission $19

The Cloisters
Suggested adult admission $25

Grey Art Gallery NYU
Suggested adult admission $3

King Manor Museum
Suggested adult admission $5

The Metropolitan Museum of Art
Suggested adult admission $25

Museum of Biblical Art
Suggested adult admission $7

Museum of the City of New York
Suggested adult admission $10; if you live or work in East Harlem above 103rd Street, visit the Museum free of charge. Simply walk in the entrance, say “I’m a neighbor,” and the suggested admission charge will be waived.

New York City Fire Museum
Suggested adult admission $7

MoMA PS1 
Suggested adult admission $10

Queens Museum of Art
Suggested adult admission $5

Staten Island Museum
Suggested adult admission $3