What is the allusion in the title of "By the Waters of Babylon"?
An allusion is a reference to something famous. It can be a poem, a piece of art, an event or in this case a location.
The allusion in this title is the name of an ancient city, Babylon. This city was known for several features. This place had magnificent culture: great foods and entertainment and great wealth. This would have been a feature city which was placed between the Tigris and Euphrates rivers. But, according to various legends and specifically the Bible, Babylon was destroyed by its own selfishness and wrecklessness.
An allusion's purpose is often to provide comparison. We have great cities that today provide rich culture. In this piece, we see the connection that a great New York might just as likely have potential to fall like Babylon did.
The allusion in the title of this story is to Psalms 137 in the Bible. You can see a link to it below.
This is a psalm in which the people of Israel are lamenting while they are being kept captive in Babylon. They are wishing that they could go back to what was once theirs.
If you think about it, John ends up like the Israelites. He finds out that there was this amazing civilization and that its people were the ancestors of John's people. He feels that his people have lost what was once theirs. He wants to get back to what his people used to have.
Now, this may be an ironic title. We are not at all sure that what John wants is going to be good for his people. He may be wishing for something that will end in disaster (they may just eventually end up like NYC already has ended up in the story).