The answer to this question can be found in the first chapter of this excellent dystopian clasic, where Randy remembers the conversation he had with his brother, Mark, and the code signal they developed together so that Mark could inform Randy if things looked as if a nuclear war was imminent. Note what the text tells us.
Mark smiled. "I won't call you up and say, 'Hey, Randy, the Russiand are about to attack us.' Phones aren't secure, and I don't think my C-in-C, or the Air Staff, would approve. But if you hear 'Alas, Babylon,' you'll know that's it."
Given the state of security and the necessity for secrecy, it is clear that Mark will have to use some kind of code langauge to communicate the perilous state of affairs with his brother. Thus they choose a common childhood memory of the nearby preacher's oft-repeated phrase in his sermons to signify the approaching terror and turmoil. It is this phrase that is of course used to give the book its title.