Amendment XV did give black people the right to vote in 1870, and some used it right away, electing state and federal senators and representatives. Jim Crow quickly snatched away that right for all practical purposes, however, as poll taxes and literacy tests--and Klansmen--denied blacks that fundamental right.atified February 3, 1870. The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any State on account of race, color, or previous condition of servitude..