Yes, it could be done.....but only as a show with a very focused megaplot.Or you could do it as a planet-of-the-week show in which the investigators are an elite squad that investigate crimes that, while rare, are non zero in something as vast as The Federation.
The Elite Cops that travel the Federation to help the local cops with the rare crimes could work. Though, all that "crime" might not feel like Star Trek.
The Federation ideal is that there would be little crime as all the technology would prevent most of it.I don’t think there being occasional crimes makes the Federation not a utopia - even in a society where everyone has enough to eat, a nice job if they want it, all the rights, and a place to live, people will still want things and people they can’t have. They’ll murder for envy, jealousy, pride, lust, anger, and even greed. Most classic golden age murder mysteries have culprits who at least appear to have enough. And I’d certainly copy the feel of those stories - things are cosy and simpatico on the surface, but people are all still people, and that means some will do the wrong thing even if they know better.
And crime really does destroy the "ideal utopia". You can't really have crime drama WITHOUT a dark, gritty crime filled society....drama needs fuel.
BUT even more then that is the utopian ideal that people just would NOT commit crimes, not matter what. That people, humanity would just: Be Good. Such people exist today.....Star Trek just has a world were it is nearly everyone.
Would a Golden Age Crime show even work? Someone does a crime....no modern drama...and they just catch the person?