Holden has good memories about visiting the museum when he was younger. The thing he likes about the place and the artifacts in it is that everything remains the same, frozen in time. This relates directly to the basis of Holden's depression. He wants to cling to the security of childhood, when things were simple and people were genuine - "natural", as it were - and not "phony". Holden cannot accept the changes that come with growing up. It is no accident, though, that when he tries to go into the museum, something inside him prevents him from doing so. Despite what Holden may wish, time marches on, and he can't go back to his life as it was before.
Holden loved going to the museum as a younger child. Based on his narrative, the constancy of the museum appealed to him—the exhibits never changed no matter how much his own life did. It’s worth noting that the changes Holden is thinking about are all negative, like kids getting a disease or parents fighting at home.
His narrative also focuses on how people would be a little “different” every time they went to the museum. He is resistant to the idea of change. This is what concerns him when he thinks about his sister Phoebe going to the museum with her classes like he did. She would be changing each time too, and that is something he does not like the thought of. His sister, who is still young, is one of his favorite people and he does not want her to change. Perhaps he is afraid that she will be subjected to the same disappointments and problems that he has been struggling with.
I kept thinking about old Phoebe going to that museum on Saturdays the way I used to. I thought how she’d see the same stuff I used to see, and how she’d be different every time she saw it . . . Certain things should stay the same.
When he gets to the museum he decides not to go in. He says “It just didn’t appeal to me.” Why? Perhaps it is because now he knows that changes are inevitable—he isn’t going to be able to stop them from happening to himself or to people he cares about, like Phoebe.