Amira Nader graduated from Columbia University in 2010 with a master’s degree in acting and nearly $190,000 in debt. She now works for a public radio station in New York City and waits tables on the side.

Ms. Nader, 31 years old, who moved to New York nine years ago from Florida, dreams of owning a home in New Orleans. But like tens of thousands of other young Americans, she is finding it hard to move away.

“I’m...