Muriel is Seymour's young wife. At first, it might be difficult to see why a man like Seymour is married to a woman like Muriel. He's spiritual and introspective; she's materialistic and social. He takes everything seriously; she's light as a feather. Everything Muriel does and says seems shallow, even more so after we watch Seymour interact with Sybil and then commit suicide. Their marriage sounds like a total mismatch.
If you really want to get a handle on this relationship, you might take a look at some of Salinger's other stories revolving around Seymour and the rest of the Glass family. "Raise High the Roofbeam, Carpenters" is a narration of Seymour and Muriel's wedding; or rather, the lack of wedding (the two elope last minute, to the dismay of Muriel's family). It features one of Seymour's diary entries and goes some way in explaining his feelings for Muriel. In a shamelessly simplified summation, she balances him out. She's sort of like his connection to the rest of the world.
Given Seymour's spirituality and wisdom, we can be pretty sure that he "gets" Muriel. But how well does Muriel understand her husband? Readers are continually struck by Muriel's cavalier indifference in the story's first few pages. Therapists think her husband is unstable. He's deliberately driven a car into a tree; he's obviously suffering mentally from his experiences in the war, and she's gabbing about fashion and giggling about his nickname for her. As it's unlikely that Muriel doesn't care about her husband, we can only conclude that she doesn't get it. She doesn't recognize the severity of the situation.
One of the enigmas of the story's ending is why Seymour chooses to kill himself in the presence of his wife, and why he doesn't wake her up before he does it. The image of the final scene is a lasting one; Muriel sleeps on one twin bed while Seymour fires a bullet through his temple. This so perfectly encapsulates the degree to which Muriel is unaware of her husband's plight. If you're interested in spirituality and Buddhism as a theme of "Bananafish," you can think about Muriel's spiritual status as compared to Seymour's. Because she's so caught up in materialistic needs, she's "asleep" and is missing what is more important.