When India abruptly banned its largest-denomination bank notes last month, New York City resident Raj Karnawat’s nearly $3,000 stash of rupees became essentially worthless, overnight.

“It’s not a small amount of money for a middle-class family to lose through no fault of our own,” said Mr. Karnawat, a citizen of India and the U.S. whose family owns a wholesale precious-stone business in Manhattan. “There was no way to be prepared for this.”

About...