Yes, alcohol should be completely banned in the European Union, because alcohol is the most abused drug. Even though making it illegal would not be popular, we were never more productive as a nation than we were in the 1920s when alcohol was completely illegal. There would be fewer deaths because of alcohol.
I do not believe alcohol should be completely banned in the European Union. It would be futile to even try. When the US instituted prohibition, all it did was open the way for speak easies, bathtub brew and moonshine. You may make alcohol illegal, but you will never eradicate it or stop it from being made and consumed in some fashion.
History has proven that prohibition doesn't work, so why do so many people want to return to it? I think it is foolish to assume that alcohol should be banned in the European Union. Banning something does not eliminate it and it tends to cause more problems than it fixes.
Banning alcohol in the European Union or anywhere else should never be banned. Sure, alcohol accounts for a lot of domestic abuse and stuff like that but if not for alcohol many of us would not even have a domestic life to abuse. Many of us would never have had the chance to breed at all and even the bible tells us to procreate.