A common argument among those opposed to suffrage was that men and women are fundamentally different and that society suffers when either sex tries to perform the duties of the other. Most arguments for suffrage did not contradict the belief in essential sex-based differences; in fact, many suffragists based their case on their belief that society needed the unique interests and abilities of women. Antisuffragists, in contrast, often depicted suffragists as women wanting to be men and questioned the manliness of the men who supported suffrage.
Note the posture and expression of the man in the cartoon. The dog says, "Humph! He's zero in masculinity." Meanwhile the woman, empowered by the right to vote for the trustees of the state university, tells the man what to do.
This link will tell you what the anti-suffragists thought about how voting could lead to the Elimination of Sex.