When out with people who are drinking/drunk, the person or people on drunk patrol are the people who are keeping them from getting into too much shit, like getting arrested or hurting themselves. Like a designated driver, but for everything.
"While I was out with everyone last night, I had to be on drunk patrol, and someone almost jumped off a bridge but I stopped them."
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"Jerry threw up last night, but I was on drunk patrol so I had to clean it up."