Starfleet already does that, at least they did in TOS. Part of the Enterprise crew's job was to patrol the frontiers of the Federation and check on colonies, mining operations, lunatic asylums and other outposts to make sure everything was kosher or investigate when things seemed squirrely and set things right.
I'd love to see that Space Sheriff aspect returned to Star Trek. It made for a good mix of stories alongside the other types. What wouldn't work is some silly procedural, since it would lock the show into an artificially constrained format.
Something like Devil in the Dark is cop-showy enough for Star Trek. Starfleet needs to have some latitude in doing their job since they never know what they'll run into next.