I don't know if UA should eliminate crime completely all by itself, but it should certainly put a massive dent into it. I mean, we're imagining future societies so technologically advanced they can imprison stars, blow up planets, convert their entire population into robots, etc. etc., this goes incredibly far beyond just a modern welfare state/UBI/living wage, it's starting out at Star Trek levels of abundance and only going up from there until we hit The Culture.
And okay, maybe some people decide they want to become crime lords, but that hinges on having things to provide the state can't, and even if e.g. drugs are illegal so what, there's going to be a million other ways to get the same entertainment or high or whatever, just go in VR Blorg Kink Sim XIV. The people who decide they would like to be Supreme Galactic Emperor are going to be no different from the same people today - cranks, crackpots, and politicians, but UA will still hamper them all because it's a lot easier to recruit a desperate underclass to your cause than pampered, well-fed, comfortably-homed, endlessly entertained people.