LEH, India—The Ladakh region of northern India is one of the world’s highest, driest inhabited places. For centuries, meltwater from winter snows in the Himalayan mountains sustained the tiny villages dotting this remote land.

Now, like many other places in India, parts of Ladakh are running short of water. A tourism boom has sent the summer population soaring, and the region’s traditional system of conserving water is breaking down.

Water...