All of which is done Redfield, in obvious cases before check-in. However as the recent problems at BA have highlighted there aren't sufficient airline or handling agent groundstaff checking-in, let alone wandering around monitoring the bars for drunks!
If the passengers are okay at check-in the bags will be checked in and labelled normally. Should they then go and get drunk in the airside bars they will not be seen until they reach the gate, and they will probably be amongst the last to do so. Their baggage hasn't been re tagged so it is just one of the many and bang goes the on time departure.
As you can see from this and other earlier responses, what action is taken depends where the passengers display signs of being drunk.