Well, for one, women didn't really want the right to vote (at least not all -- anti-suffragettes were not in favor). One of the major reasons that they did not want the right to vote is that it could bring them into conscription or other war-related activities (which anti-suffragettes were not particularly fond of). Domestic life also seemed more amiable for a lot of women, who were not particularly interested in the state of politics.
As for general anti-arguments, you have the fact that many people did not believe that women would be as knowledgeable on voting/politics as men (also why rich and high-class women were given the right to vote before those who were of a poorer class).