Ms. Gillibrand has sought to position herself as the advocate-in-chief for women. Her leadership on abortion fits with her broader strategy.

Her campaign has struggled to raise money from small donors, but the recent anti-abortion legislation passed in numerous states helped give Ms. Gillibrand a bump. She booked a last-minute trip to Georgia to decry the passage of a restrictive abortion law there, blitzed the cable news networks, and the donations that followed helped secure her spot in the June debates.