Third Person (Omniscient); but also First Person (Peripheral Narrator)
If you look only at the text of "Bananafish," you see an omniscient third person narrator. It might as well be a fly on the wall telling the story – the narrator doesn't know anything about these characters other than what he sees. Notice that the narrator never tells us that Seymour just came back from the war; instead, he observes Muriel and her mother discussing the fact that Seymour just came back from the war. Similarly, Muriel is referred to as "the young woman" and Seymour as "the young man" in narration.
But, check out more of Salinger's work on the Glass family, and you'll meet Buddy Glass, Seymour's younger brother and a writer. He claims to be the "hidden" narrator of many (or maybe all) of the Glass stories. At one point there's even the suggestion that Buddy, though trying to describe Seymour's death, unintentionally ends up describing himself in the character named "Seymour." There's plenty more narrative trickiness where that came from: check out Franny and Zooey if you find this sort of thing as interesting as we do.