CÚCUTA, Colombia—Hundreds of thousands of Venezuelans are fleeing their country’s misery and pouring across borders into nearby countries, particularly Colombia, creating a sharpening challenge for the region.

As the collapse of Venezuela’s economy deepens, the number of those fleeing is accelerating. Nearly 3 million Venezuelans—a 10th of the population—have left the oil-rich country over the past two decades of leftist rule. Almost half that number—some 1.2 million people—have gone in the past two years, according to Tomás...