Seymour holds a number of Glass family superlatives: he was the oldest, "the most intricately calibrated," and "the best" and "most rewarding" to hear on "It's a Wise Child" (Zooey.4.73, Z.2.3). Buddy informs us rather clinically, via footnote, that in 1955 Seymour has been dead almost seven years, having killed himself while vacationing in Florida with his wife. Seymour, along with Buddy, was responsible for Franny and Zooey's spiritual education. We gather from the little bit we hear of him that he was particularly spiritual himself; the Glass children seem to revere him as a sort of religious guide.
Salinger devoted quite a bit of time to developing Seymour's character – it's just that we have to go outside of Franny and Zooey to find it. The short stories "A Perfect Day for Bananafish," "Raise High the Roof Beam, Carpenters," and "Seymour: an Introduction" explore his character in length.
We find out that Seymour was as meditative and spiritual in these other stories as suggested here in Franny and Zooey. He was also a poet, and definitely an oddball; he seemed to get along better with children than adults. Seymour shared his siblings' disdain for materialism. His suicide may have been caused by a sort of post-traumatic stress disorder he suffered after returning from the war.
But back to Franny and Zooey, at least for the time being (though you should definitely read Salinger's other Glass stories.) Zooey is right to point out that the "whole goddam house stinks of ghosts," one dead ghost (Seymour) and one half-dead ghost (Buddy) (Zooey.5.53). Seymour's death really does hang over the Glass house; it hangs on Bessie, it has a huge effect on Zooey, and we can only imagine (or read in other texts) what it has done to Buddy.
Seymour's presence is tragic, but there's also a sort of poetic beauty to his influence – after death – on his younger siblings. Seymour continues to teach his brothers and sisters, and still in his own cryptic way. His lesson of the "Fat Lady" is passed on to Franny through Zooey's lecture. His two books made their way into Franny's hands to begin with. His diary entry spurs Zooey to make the fake phone call that leads to Franny's resolution. Even Seymour's poetry makes it into the text, via Buddy's letter. For a man who's been dead for several years, Seymour certainly has a lot to say in this novel.