Bernie Sanders is 74. He grew up playing stickball in the streets of Brooklyn and watching a black-and-white television. Yet this child of the 1940s, who says Franklin D. Roosevelt is his favorite president, has inspired a potent political movement among young people today. Young voters' support for Mr. Sanders has created a quandary in Hillary Clinton's campaign headquarters in Brooklyn, where millennial staff members have tried to persuade their peers to back the former first lady. The discomfort, and, in some cases hostility, toward Mrs. Clinton among young voters is striking. Some of them, feeling the pinch of economic...