http://pricetheory.uchicago.edu/levitt/Papers/LevittUsingElectoralCycles1997.pdf

This paper is pretty much the gold standard in the field because it makes a very clear identification of the effect of police. Here is a quick summary of it.

Abstract: Previous empirical studies have uncovered little evidence that police reduce crime, possibly due to simultaneity problems. This paper uses the timing of mayoral and gubernatorial elections as an instrumental variable to identify a causal effect of police on crime. Increases in the size of police forces are shown to be disproportionately concentrated in mayoral and gubernatorial election years. Increases in police are shown to substantially reduce violent crime, but have a smaller impact on property crime. The null hypothesis that the marginal social benefit of reduced crime equals the costs of hiring additional police cannot be rejected.


And another quick summary of the literature in this area.

http://marginalrevolution.com/marginalrevolution/2012/12/police-crime-and-the-usefulness-of-economics.html