I'm really not an expert on this, but its my understanding the first wave of feminists were very tightly connected with the temperance movement, and that the two amendments, prohibition and women's suffrage were passed within a year of each other. Obviously suffrage was a no brainier, but weren't there other worthwhile causes to replace temperance. Was there a debate in the movement about different goals? I don't want to move into historical what if territory, but it seems like if these movements had the clout to pass two amendments we would be better off now if they had found a different cause, like universal healthcare for instance.
The assumption I am going to make from the context of your question is you are talking about feminism in the US. Second, since you are talking about Prohibition, I have to assume that you discount the Seneca Falls convention of 1848 and the works of Elizabeth Cady Stanton. (Not to mention the female philisophs of the enlightenment like Mary Wollstonecraft, or the founding mothers like Abigail Adams)
The answer to your connection is the structure of middle and upper class american society following the American Revolution. The idea of separate spheres of influence for men and women emerged. Men went out into the world and worked, women stayed at home and guarded the moral purity of her husband and children. The idea of women as moral guardians emerged in the middle and upper class ranks (poor women were forced to go out and find work to sustain their families, this includes black women, both free and slave).
Because drinking gets men drunk, and because drinking costs money, and because drunk people do stupid, immoral things like gambling, wife beating, buying magic beans, etc, many women took this idea of being the moral guardian to mean they must squash out immorality outside of the home. A man cannot be properly moral if he is drunk. A woman cannot protect the morality of her children if she has two black eyes.
Therefor a major aspect of women guarding the morality of the home and the family was to eradicate various evils, Drinking was one of them, Slavery was another.
Because women became the guardians of morality many women agitated more and more for the right to vote so that they could help elect morally proper men to office. (but not all, in fact half of women refused to vote after they were granted the right to, first state by state, then nationally, and this impacted voter turnout percentages). Women needed to vote to protect the morality of men against themselves.