NEW YORK—Elizabeth Warren cemented a fight against corruption as the argument underpinning her presidential campaign, telling thousands gathered in this liberal bastion that “giant corporations have bought off our government.”

Earlier on Monday, Ms. Warren released an anticorruption proposal on the blog site Medium that built on legislation she announced last year. It adds new lobbying restrictions including a federal ban on all fundraising activities hosted by lobbyists. But the evening speech at Washington Square Park in...