Think the original idea for releasing this information was to prevent the police from secretly disappearing people, which has happened quite a bit around the world. The police know, if they take someone into custody, that it will be part of the public record for all to see.
Which is fine when it's just the police making that information available on a short-term, need-to-know basis. When you have third parties (like "mugshot sites") republishing that information in a more lasting and public form, it becomes a problem.
We really need to limit the reasons a person can be taken into custody. There are valid reasons, and we need to evaluate those reasons too. Too often, people just end up in custody (and then depriving the public of their usefulness).