Last spring, Cecile Richards’s BlackBerry buzzed with an unexpected text message. It was from her son Daniel, a college student in Pennsylvania. He was heading off to Toledo, having organized a bus trip of friends to attend a rally supporting Planned Parenthood. The message came as Congress was debating ending the group’s nearly $100 million in federal funding.

Richards was surprised: Despite her five years now as president of Planned Parenthood, her son had never been active in abortion politics. To her, Daniel and his friends represented a wave of young supporters whom groups such as hers had long struggled to engage. All it took was a sustained attack on government funding of family planning, waged at the federal and state level, to get...