As the article notes, at least in the UK, taxes in the 50's were three times as high. As someone who has distilled alcohol in a garage before, it's incredibly labor intensive at small scales. Brewing beer is doable, but it is not "extremely easy" by any measure, especially if the goal is reasonably decent tasting beer. Alcohol taxes would need to be extremely high before black market alcohol became an issue of any significance.
Brewing beer for example is extremely easy. Everybody can do it and there is even a whole community around brewing at home.