Roddenberry had a negative influence on the later Star Trek (TNG, DS9, VOY). His vision of an Earth as a perfect paradise and where humans don't have interpersonal conflicts lead to stories in which conflict had to be introduced from an outside agency. This created a nightmare for the writers. Roddenberry's ideology had poisoned his creation. JJ Abrams has brought Star Trek back to its TOS roots, where Earth isn't a paradise and humans have interpersonal conflicts.
Don't most of the writers for TNG admit the 'Roddenberry box' is what forced them to be more creative with their stories?
I don't think the lack of crime on Earth is meant to be an absolute statement. Crime must exist, but it's unexpected to the point that most people feel safe enough to leave their door unlocked. There's obviously crimes of passion, and the person here and there who tries to get around trade regulations and such.
But, in a world where scarcity is eliminated, it's easy to imagine there's very little crime.