The bananafish are one of the story's key symbols. To understand what's going on here, we've got to take a closer look at the text:
"This is a perfect day for bananafish. […] Their habits are very peculiar. […] They lead a very tragic life. […] You know what they do, Sybil? […] Well, they swim into a hole where there's a lot of bananas. They're very ordinary-looking fish when they swim in. But once they get in, they behave like pigs. Why, I've known some bananafish to swim into a banana hole and eat as many as seventy-eight bananas. […] Naturally, after that they're so fat they can't get out of the hole again. Can't fit through the door."
[…] "What happens to them?"
"Oh, you mean after they eat so many bananas they can't get out of the banana hole? […] Well, I hate to tell you, Sybil. They die. […] They get banana fever. It's a terrible disease." (2.71-83)
As with most of "Bananafish," there's no one answer or clear interpretation here. One angle you might take is to think about the story's spiritual or Zen Buddhism theme. (See "What's Up with the Epigraph?" for an introduction to this theme.) By stuffing themselves full of bananas, the bananafish are focusing physical needs or pleasures. This is not unlike the materialistic adults in the story (such as Muriel, Muriel's mother, and Mrs. Carpenter) with their talk of clothes, fashion, or drinks. Seymour, who sees more, is aware of this sort of gluttony and wants to avoid it all costs. He doesn't want to gorge himself on bananas.
Which leads us nicely into the discussion of "What's Up with the Ending?" If Seymour is enlightened, then killing himself is a way of triumphing over any material impulses. If Seymour is filled with shame at his death, it may be that he suspects himself of such "banana-fever." (Maybe when Sybil exclaims that she's seen a bananafish, Seymour thinks he's talking about him.) Go ahead and check out "What's Up with the Ending?" for more thoughts.