"Children of Men" (2006) tries to tackle the question, and in general does a pretty good job. Credibility however suffers from British home bias. It's hard to believe that the UK would have remained in "relatively good" shape so many years after infertility had set in, and the idea implicit in the story that the US would become some kind of violent and chaotic no man's land, apparently due to access to guns and military equipment, is inconsistent with history and pretty naive. On the other hand, Mexican-born director Alfonso Cuarón does an excellent job making the UK look like a third world country.
It appeared to me that the movie also suggests that some "remedievalization" of the UK takes place through nobility, what would have helped the UK to remain as some kind of semi-fascist, semi-feudal state. I doubt however that feudal and fascist institutions would prosper without "heirs to the throne."
Despite my skepticism regarding some of its assumptions, I must say that it's an interesting and well-done film. Watch it however only if you have a taste for futuristic dystopias involving government distribution of free suicide kits and mandatory fertility exams... Here's the trailer: