In my state bartenders are legally obligated to not serve "visibly drunk" patrons. Though only the nicer bars actually follow this rule, and it is more in place so they can easily boot out unruly drunks or bar entry for people that are already wasted before they show up.
A robot bartender in a commercial environment would either need to be able to follow all the same rules or be operated by someone that does.
The question is... If you are in your own home, does the robot count as a bartender, or is it an appliance? My guess is the latter, the responsibility belongs to the operator.
Though it would be amusing to see the door to the refrigerator refuse to open for a drunk person.
"I'm sorry Dave, I think you have already had enough to drink."
"Hey buddy, can you come in to my house and open my fridge for me?"