The claim for publication is that the public has a right to know about potential criminal activity and so on. Of course, the reality is that publication of the mugshot is accepted by many as proof of criminal liability notwithstanding disclaimers to the contrary.

It's less clear why police departments publish them, but I think it has a lot to do with sheriffs and prosecutors being elected offices in the US. I've never agreed with this process since the potential for abuse is so obvious, and also since it makes it that much more difficult, time-consuming, and expensive to assemble a neutral jury when a case does go to trial.