It may not seem like an apropos mix, but bear with me. After all, one can prohibit just about anything. Even in actual Prohibition, it could be argued what the actual thing intended to be prohibited was: a substance (alcohol), a state (drunkeness) or an activity (carousing).
But not to stray too far from either source material, I’ll assume that what’s prohibited at Hogwarts (and allied venues) are certain kinds of alchemical potions.
So, when Prohibition hit, a formerly legal activity had become illegal overnight. This wasn’t good for people who liked a drink, but they were at the bottom of that particular economic pyramid. Above them were the sellers, the distributors, the makers, and all those primary producers who supported the other stages: farmers, bottle-makers – even the writers of drinking songs.
When certain types of potion are out-lawed at Hogwarts (probably after some sort of abuse scandal), the first thing that happens is that one Severus Snape is made profoundly unhappy. Whatever his other motivations may be, he’s a teacher and maker of potions, and to have his field of endeavour suddenly made smaller can’t make him happy. (Not to mention all the fun of retrieving a whole lot of now criminal textbooks.)
But he’s well-connected in Slytherin circles, and let’s face it, if someone decides to become an alchemical bootlegger, we all know what house they’re going to be from. Certain stores in Diagon Alley may develop back rooms that can be entered only if you know the right passphrase, and no doubt it will spread from there to Hogsmeade and other such places that live in the shadow of magic.
What this means, if events follow the same progression they did during Prohibition – and there’s no reason to think they wouldn’t – is that slowly but surely, Slytherin House will develop more power and wealth. Meanwhile, Prohibition on potions regarded by most as innocent erodes respect for law and authority, weakening the other houses in a less tangible but no less real manner.
And when different factions within the Slytherin ranks break out into the magical equivalent of a gang war, well, then you’ve got yourself a game!
Of course, the most probably outcome is that, as in our own reality, Prohibition will be recognised as a failure and repealed – but not before the empowerment of criminals and evil-doers has progressed to a point that is almost impossible to roll back to the previous status quo…