In John Cheever's "The Swimmer," I don't know if you are supposed to find a specific answer, as much as look to his experiences as a metaphor of his life.
Neddy comes from the upper echelons of society: he and his friends are wealthy with beautiful homes, servants, and free time to idle by the pool, party and enjoy their lives.
When he decides to swim home, I take this as a metaphor for his past: he sees people he knows who welcome him and are glad to see him. He continues as he stops at each home to feel strong and self-assured.
However, then the storm hits. This is symbolic of tragedy striking. It could be any number of things: losing his wealth in the stock market; running through his money; investing in unwise business deals. The cause is never clear, but the mood of the story changes dramatically here.
He visits one family that speaks to his "misfortunes," things he was not aware of. What is the cause? It could be alcoholism, as you suggest. There is certainly a great deal of discussion about drinking: the story begins with everyone saying how they had too much to drink the night before. The fact that this seems to be a joke at the beginning may speak more to the lifestyle they lead and less to Neddy's particular problems, but there is no way to know. He certainly feels the need for a drink more often after the storm, believing that this will strengthen him and make him feel better. There is enough supporting detail to make an argument for this.
It could be some kind of mental or emotional breakdown where he refuses to face the new reality of his life: he has lost everything, it seems: his house has been put up for sale; he has visited several people, including his old mistress, to borrow money. The hostess that is so rude talks about him to other guests in his hearing about his "fall from grace." It could be that he is simply repressing these details from within a serious depressive state.
Regardless of the cause, by the end the reader is aware that the journey Neddy has been on has taken him from the heights of social acceptance to the brink of ruin, and he seems confused as to what has happened. Either alcoholism or some kind of breakdown could be responsible. It is up to the reader to decide which situation is most clearly supported with the author's details.