Dying crops, spiking energy bills, showers once a week. In South America, the climate future has arrived.

Sergio Koci’s sunflower farm in the lowlands of northern Argentina has survived decades of political upheaval, runaway inflation and the coronavirus outbreak. But as a series of historic droughts deadens vast expanses of South America, he fears a worsening water crisis could do what other calamities couldn’t: Bust his third-generation agribusiness.

“When you have one bad year, you can face it,” Koci said. Some of his 20,000 acres rest near the mighty Paraná River, where water levels have reached lows not seen since 1944. On the back of two years of drought-related...