There's a bit of truth in it. Some of the same women asking for the right to vote were also actively trying to shut down saloons. Alcohol and alcoholism were seen as a threat to the family and the home- the domain of women. Alcoholism was mostly a male disease: men were allowed into saloons, where they could drink up their wages, women could not- but they and their children would suffer the consequences. Both for Susan B Anthony and Elizabeth Cady Staunton, a great early motivator for their drive to get votes for women was the seeming impossibility of banning the sale of alcohol without it- their protests were too easily ignored. Because of this, prominent temperance groups , like the Women's Christian Temperance Union, would continue to advocate votes for women.

However, while there was a very strong element of women's rights in the 19th c. Temperance organizations, and those organizations were a powerful political force, the Volstead Act and the 18th Amendment to the Constitution were actually achieved before women got the vote. Single-issue advocacy was pioneered by Wayne B Wheeler, the immensely energetic lawyer who founded the Anti-Saloon League. He even coined the term "pressure group". Wheeler confronted politicians with a stark choice: vote for Prohibition, or they and other Temperance groups would back another candidate at the next election. With the threat of Temperance movements campaigning, marching against them, ( and with no organized resistance from distillers, breweries) most politicians fell into line and the Amendment was passed.

Okrent:Last Call