Public drunkenness was a minor criminal offense of being intoxicated in a public place. Those who committed this offense were liable to be arrested and sent to the brig.
In 2369, a Bajoran named Kainon was living on Deep Space 9 when he was detained for public drunkenness by Constable Odo. He was subsequently placed into a holding area facing the one containing the Cardassian Aamin Marritza. (DS9: "Duet")
In 2370, Odo and Quark arrested Kor after the latter was found drunk inside one of the Holosuites playing the Battle of Klach D'kel Brakt. Quark shut off the power and Odo arrested Kor and placed him in a cell to sleep it off. Koloth came by later to bail Kor out, but when he saw Kor was still drunk, he told Odo to keep him and stormed off to the Replimat for lunch. Dax bails Kor out and the two of them leave to join Koloth at the replimat. (DS9: "Blood Oath")