If there's one fact that Americans take for granted, it's that other people want to live here. As President Barack Obama noted in his speech on immigration earlier this week, the U.S. has always attracted strivers from every corner of the globe, often willing to risk great hardships to get here.

During the 20th century especially, America became a magnet for the bright and ambitious. Millions of talented foreigners, from Alfred Hitchcock to Sergey Brin, flocked to our universities...