I think that touches on an issue which affects us even today. How much responsibility do we place on the human individual's decision to partake in evil and how much do we place on the object which helps to facilitate evil? It's the same issue with guns; some want gun control to prevent crime, others want to control criminals to prevent crime.
Because alcohol is addictive, I can see how Carrie Nation and others could have found this to be the end against which their crusade was aimed. As I've stated in another thread, cocaine was discovered to have a very detrimental and addictive effect on people in the early 20th Century. Other drugs probably underwent the same scrutiny around this time. It's important to take all of this into account when we ask why alcohol was prohibited. If we consider that liquor was outlawed in a vacuum it might seem odd, but if we consider that it was prohibited along with cocaine, perhaps heroin, opium, etc, all of which were seen to have had a demonstrably evil effect on Americans and their family life, it's a different story.
It then took some time for the public to separate the evils of drugs from the evils of alcohol and to realize they weren't quite equivalent.