Let's get away from this mortality business and talk instead about the inevitable passing of time and the changes that it brings. For Holden especially, this is a source of depression; he doesn't like that everything has to change, that everyone has to grow up. "Certain things […] you ought to be able to stick […] in one of those big glass cases and just leave them alone," he says. He's making a rather explicit connection here between the Indian Room at the museum (where the displays always stay the same) and the children (who are always changing) who visit on field trips. Since Holden is so straightforward about the connection, you'll really a lot from reading what he has to say on the subject (the last four paragraphs of Chapter Sixteen).
Yet, there is one interesting (and less straightforward) aspect we'd like to at least address. Holden says that while the displays stay the same, a person is different every time he comes back to visit. But he makes the point that it's not so much about getting older as it is about becoming different. Before we read this, we might have been tempted to make the connection that, since Holden fears dying, and he also fears getting older, the latter is simply a fear of getting old. But this passage here prevents us from drawing that conclusion. Holden emphasizes that it's not aging that bothers him – it's the changes one goes through in order to become an adult. So he's talking more about the intangible qualities of youth and innocence than he is about the physical ones. Just look at his list of examples – hearing parents fight, or seeing a gasoline puddle – these are examples of awareness, of mental growth, not of physical aging.