BOLIVAR, Pa.—Sen. Bernie Sanders calls himself a democratic socialist, and when voters scan the Web to see what that means, the trail often leads to a band of about 6,500 leftists pining for relevance in a country that has long deemed “socialism” a dirty word.

Since he entered the race in May, the Democratic Socialists of America have seen monthly new members spiking two and three times the levels of a year before.

Nearly...