The Crime Detection System will either operate with remote tags or from you hacking phones from the citizens of Chicago you will pass by. Turn on Profiling using SQUARE and then cycle through the pedestrians to come across premeditated crimes, ATM accounts, songs, System Keys, vehicles, or even gang hideouts/criminal convoy locations. Crimes will appear on your mini map (and the main map) as upside-down blue triangles. What you have to do is stalk/tail the victim or the criminal to watch for the crime to take place. Don't idly stand by and watch the crime continue. Once the crime's % chance starts escalating, walk (don't run) as close as you can to the criminal without alerting it to get ready.
Running towards the criminal will alert them to your presence, sending them running prematurely (and ending the crime). If you run after them right as the ctOS alert comes up suggesting you to Intervene, the criminal will start to run away. Either at this time or right before s/he starts to run, you can take care of them with a non-lethal takedown (CIRCLE) to take care of them quickly and earning you positive reputation.