I always thought the solution was simple - unlike the printed page, web articles don't have to be static. The publishers should allow the subjects of these articles to provide updates to the beat reports (subject to verification of course) to document the disposition of the arrest. You could imagine an article might say "June 1, 2003: Jane Doe arrested for drug positions (insert mugshot/arrest record" followed up "Update Feb 1, 2006: Jane Doe indicated that the case was dismissed, which was verified by court records. Ms. Doe has not been arrested since and documents provided by Ms. Doe (and authenticated by this publication) indicate that she graduated with honors from XX college and has been successfully employed as a mechanical engineer to positive reviews."
That may not help with the extortion racket problem with the mugshot sites, but one would hope that the media outlet websites would rank higher than the mugshot sites on search engines.