Some of the storyline missions give you a reputation bonus for completing them with minimal bloodshed. But considering that these aren't repeatable, they are not a sustainable way to fix your reputation.
The best way to do so is to prevent crimes. Note that you get a greater reputation bonus for both taking down the criminals before they hurt the victims (but not before the "Crime propability" bar turns red and the "Suggestion: Intervene" HUD message appears - you won't get any reputation bonus when you scare away the criminal too early) and for performing a melee takedown on the criminal instead of just killing them. There are three different ways to notice crimes:
- Remote profiling crime reports you get automatically while being in an area with unlocked ctOS connectivity
- When you profile pedestrians you will sometimes find people with a "potential crime in progress" description. Scanning these people will also trigger a crime report.
- Sometimes people will perform spontaneous roberies on the street. You notice this by an enemy-icon suddenly appearing on the minimap
You are more likely to find the latter two in areas with a high density of moving pedestrians, like parks or the promenades near the water. The pedestrian-density seems to be higher during the day than at night. Also keep in mind that none of these events happen when you are currently on a mission.