How does death serve as a theme throughout the story?

The theme of death appears in the very first line of The Brooklyn Follies:

I was looking for a quiet place to die. Someone recommended Brooklyn, and so the next morning I traveled down there from Westchester to scope out the terrain.

The narrator of the story, Nathan Glass, is going through numerous upheavals in his life. As well as his recent divorce, he's taken early retirement from his career in life insurance. Most seriously of all, he's in remission from lung cancer, which has had a devastating impact on him, both physically and mentally. It seems that Nathan has lost everything: his wife, his job, his will, his hair. As the story begins, it's an open question as to how much longer Nathan can go on.
But here as elsewhere in Auster's work, death, or the contemplation of death, is presented as an opportunity for renewal, the catalyst for a major life change, an existential rebirth. Nathan channels his depression and preoccupation with death into something positive and creative. He's going to write a book of human folly detailing "every blunder, every pratfall, every embarrassment, every idiocy, every foible, and every inane act" that he and others have committed in their lives.
In embarking upon this project, Nathan is effectively putting his old self to death—the Nathan Glass held back by an unhappy marriage, an unfulfilling career, and the ravages of cancer. In The Brooklyn Follies death is presented by Auster in starkly existential terms; it forces us to confront the absurdity of life and make a decision as to how we will respond. Nathan has made his choice. He's going to stare death right in the face and counter it with a detailed compilation of his, and humankind's, folly.