People, mainly men, opposed Women's Suffrage because they felt that it was impractical for a woman to have the same rights as men. During this time, men were considered superior beings to women and that the woman's job was the bringing up of children and caring of the house, not deciding who should govern the citizens. Obviously, they were chauvinist pigs who weren't capable of realizing that everyone is equal.
(Sorry for the tidbit at the end, that's how I feel personally, but you can change it as you please.)