Who supported Prohibition and who opposed it?
Women and Protestants liked it, men and Catholics did not.
It's no accident that both prohibition and the women's vote got passed while a huge number of male voters were away fighting World War I (although neither took effect until just after the war ended).
Women's Suffrage groups liked the idea of prohibition because a lot of men were drunken alcoholic brutes to their wives. Protestants liked prohibition because heavy drinking was commonly associated with Catholic Irish, Italian and German immigrants from over the previous fifty years. If more working class men had been in the country to vote against politicians who lobbied for prohibition, it likely would not have passed.
But then, neither would Women's Suffrage. You have to take the good with the bad.