Not a good avenue of regulation. It doesn't matter to me if somebody gets drunk and carries a weapon, it matters more if they have a history of crime, substance abuse, mental instability, or domestic violence. Background checks are more important. This is also a difficult idea to enforce.
If not, what penalty should be given for such act? Should it be punishable to the extent that drunk driving is? How exactly should it be regulated?
When you are drunk or drinking, you have a diminished capacity. I am not for more laws because they do little to dissuade those who do not have common sense who will do those things anyways; look at all the people who still have cell phones glued to their faces while driving. In any case, if a firearm is used and the shooting is unjustified, the person will already be punished for it whether drunk or sober.
Even rights have responsibilities and if you are drunk in a state where it is legal to carry a fire arm, you waive that right until you are sober.
Its sad that there even needs to be a law for that which should be common sense.