Ah, post apocalyptic literature. Normally, that's a genre I tend to like, especially because most of it usually has a clear cut good guy/bad guy. That is not the case for Stephen Vincet Benet's "By the Waters of Babylon." The protagonist is the narrator of the story. He is the person most deeply involved in the "action." I say "action," because not a whole lot happens in the story.
As for the antagonist, you might make a case that it is nature. Obviously the city and social structure that you know today don't exist. Nature is taking back over, and I'm sure the narrator is at risk from predators. However, that feeling isn't clearly emphasized in the story. I would say the main antagonist is ignorance. It's not just the narrator's ignorance, though, but his "people's" ignorance as well. They put a lot of faith in mysticism. They are forbidden to go East, forbidden to go to the place of the gods, forbidden to touch certain metals, etc. No explanation is given as to why, and I think it is because the people and narrator simply do not know why they believe what they believe. They don't know the events of the past. They don't know what those tall buildings and bridges are that go to New York. Society has completely forgotten all of the learning and knowledge that had accumulated for thousands of years before them. They have essentially been sent back to the Stone Age because of lost learning and knowledge. That is why I think ignorance is the antagonist of the story.