Does immigration enforcement reduce crime?

No.  From Thomas J. Miles and Adam B. Cox in the JLE:

Prior research investigates whether immigrants commit more crimes than native-born people. Yet the central policy used to regulate immigration — detention and deportation — has received little empirical evaluation. This article studies a recent policy innovation called Secure Communities. This program permits the federal government to check the immigration status of every person arrested by local police and to take the arrestee into federal custody promptly for deportation proceedings. Since its launch, the program has led to a quarter of a million detentions. We utilize the staggered rollout of the program across the country to obtain differences-in-differences estimates of its impact on crime rates. We also use unique counts of the detainees from each county and month to estimate the elasticity of crime with respect to confined immigrants. The results show that the Secure Communities program has had no observable effect on the overall crime rate.

That is once again via the excellent Kevin Lewis.