Which New York museum does Holden visit because he likes that everything stays the same?

The museum Holden visits because he likes that everything stays the same is the Museum of Natural History, formally the American Museum of Natural History. In Chapter 16, he walks there while remembering school trips from his childhood.

Holden is fascinated by the museum's displays because the figures and animals never move or grow older. He recalls that the Eskimo, birds, deer, and other exhibits remain fixed in exactly the same positions every time he returns. The museum therefore represents the stability and permanence he wishes life could offer.

There is an important detail often missed: Holden walks to the museum but ultimately decides not to go inside. He realizes that the museum itself may be unchanged, but he is not. This contrasts with the Metropolitan Museum of Art, where Holden later meets Sally Hayes; the two museums are separate locations in the novel.