Mrs. Clinton began her day in Philadelphia, speaking at the Mount Airy Church of God in Christ with Senator Cory Booker of New Jersey, a fellow Democrat. In a city where high black turnout could seal her grip on the presidency, Mrs. Clinton framed the campaign in the context of historic struggles for equality — from the origins of the women’s rights movement in Seneca Falls, N.Y., to the march for black civil rights in Selma, Ala.