For more than a decade, the youth-and-civics group Rock the Vote has been the coolest kid on the political playground. Founded in Los Angeles in 1990 with the goal of politically empowering the MTV generation, Rock the Vote quickly became a cause celebre among Democratic and entertainment power brokers. At rock concerts, on college campuses and with ads featuring a near-naked Madonna, the group helped register millions of young voters. But as it moves into its 16th year, Rock the Vote itself is being rocked by crisis. Saddled with about $700,000 in debt, the group has cut its staff from...