How does the setting affect the mood of the story and what is the story's theme? (By the Waters of Babylon)

I think there are two major themes in this story.

First, there is the theme of humanity's ability to destroy itself.  In the story, New York has been destroyed, presumably by nuclear war.  This shows how we are able to use our own abilities in very bad ways.

Second, the story shows human resiliency.   Although New York has been destroyed, John and his tribe have kept some amount of learning alive. At the end of the story, John vows to rebuild.  This shows the strength of the human spirit.

In my opinion, the setting makes the mood quite dark.  The burned out city with its packs of feral dogs is a very gloomy setting.