You may be interested in this study: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=586405

The authors used changes in the terror alert level (which caused increases in police presence) as a natural experiment, and found that crime dropped significantly in the area when the terror level increased.

I think the explanation here is probably quite simple: an increase in the number of police officers in an area means that calls are responded to more quickly and for any given crime the criminal has a higher chance of being arrested. As we'd expect, a high likelihood of negative consequences deters crime.