How do you handle a drunken abusive customer on the phone?

A person in that state will be not be open to reason. Arguing back and forth will get nothing accomplished. Just say that although you would love to help them, you are unable to help them at this time, eg. "We may have a bad connection at the moment, so I'm having trouble understanding what's being said" . Ask for their phone number and say you will have someone call back tomorrow when they can devote the proper attention to this matter. I would keep my tone sympathetic, because I understand they are mentally challenged at that moment.

You can't assume they are drunk. Ingore that part of them. Just tell them you'd be really happy to help them, but they have to be considerate of you and stop talking that way. If they continue give them another warning and tell them you will have to hang up if they continue to act that way. Maybe one more warning if you feel necesarry. Then if they act abusive again, tell them you are hanging up because you don't appreciate the way they are acting and if they call back and are considerate someone will be happy to help them and hang up. Then report it to your supervisor.

Just let him or her go on until they get bored.

Hang up.