The waterway’s decline poses an existential threat to Elephant Butte, N.M., and surrounding farms. ‘This place is going to turn into a ghost town.’ ELEPHANT BUTTE, N.M.—Three decades ago, a rare event drew hundreds of people from miles away to the Elephant Butte reservoir, this state’s largest lake. The reservoir, which supplied drinking water to thousands and nourished vast acres of farmland, had swelled from the Rio Grande and begun to overflow. Jim Paxon remembers the roar as water cascaded down the dam’s spillway, tossing mist into the air and churning the ground below. Those days are long gone. One...