This comes up a lot. While it works as an explanation for the US, it's an international issue; it's not one of those things where the US does it differently without realising.

What happens is usually a mix of factors. Drugs aren't purely harmless, you can certainly find a few people who've been harmed directly and a larger group of people whose use is problematic. This creates a demand for Something To Be done.

This falls down on comparisons with alcohol and tobacco - both of which are widely legal and have their own temperance movements, but haven't quite succeeded in a total ban yet. America (and a few other places) actually did have a War On Booze, they just gave up because it's too socially normalised. Only the more extreme Islamic countries manage official temperance, and even there it's widely violated.