Part of the reason for that is that the rest of the world was so incredibly poor that only a handful of refugees could make it to the US. There was simply a lot less migration and asylum-seeking back then. It wasn't because people were better off. It's really because they were worse off.

The rest of the world is now richer, and most of the world can afford access to a cell phone. That means they know about the US and how much better it is there, and can often find a way to try to slip through the border to get refugee status.