Well this is very interesting. Is there some kind of "zero threshold" under which nothing ever visibly happens, maybe? How long does a "growing" reputation pool last before expiring? Is there a roll to see whether growing moves to established, and if so what gets rolled?
As far as ways to reduce the rep - one would assume that spending time in public in the company of esteemed members of the community, and probably in particular law enforcement, should drop it. Talking to the fuzz extensively should really for all intents and purposes tank one's rep amongst other criminals but that's probably a separate rep tbh. This one just seems to be for the purpose of one's publicly perceived by the average citizen reputation, which is a very different thing from one's reputation as a criminal amongst one's fellow criminals. (Meta reputations, anyone? Ooooh)
Doing charity work and/or donating money to the church/whatever religious institution is considered legitimate and upstanding might be another pathway, again to be done in a public-facing manner in order to have an effect on rep obvs. Might be a case where you could implement tasks like for jobs, but available to anyone.
Finally, maybe just a lifestyle/behaviour option as far as how hard one is working to improve or decrease any given reputation in the background, again similar to the job effort? This might allow for some small wiggle room such that someone who is actively trying to get in good with the locals and be perceived as upstanding can differentiate themselves from someone who does not care/wants to be seen as a Bad Guy.
Edit: I've had another thought on this last one - in order to make it so that you can't just get infinite free credit with no effort just by setting yourself to be virtually working on every reputation ever at 100% effort, probably this should be limited to one reputation at a time? So something like "lifestyle reputation decrease criminal 100%" to focus your offline/background/downtime efforts 100% on lowering your criminal reputation, but you can't do this AND ALSO be lowering your "foreigner" rep, or raising your "local" rep, or whatever else! Also potentially you could get into territory where putting "100%" effort into too many things eventually means that you cannot, in fact, put that much effort in and it gets split/divided across all aspects with all suffering...
(No, no, I haven't thought extensively about reputation before, why do you ask? 🤣)