Originally, The Clink was the name of an English prison that operated from the 12th to 18th centuries. Apparently, the name derived from the sound of the shackles being hammered into place on the prisoner's ankles and wrists. (No such thing as handcuffs, in those days!) The name of the prison morphed into the phrase "in the clink." You don't want to be there, because you'd be in prison.
From the novel All Quiet on the Western Front:
One Sunday as Kropp and I were lugging a latrine-bucket on a pole across the barrack-yard, Himmelstoss came by, all polished up and spry for going out. … In spite of ourselves we tripped and emptied the bucket over his legs. He raved, but the limit had been reached.
"That means clink," [Himmelstoss ] yelled.