The Marshall Project, an independent nonprofit news organization devoted to researching crime and the criminal justice system, has released an interactive report on line examining the relationship between immigration (both legal and illegal) and crimes such as robbery, murder and assault. They assembled data from 200 U.S. cities and for the nation as a whole, and they provide graphs of immigration and crime rates that you can review for yourself here.
In brief, an overall increase in the number of immigrants since the 1980s has been accompanied by a drop in crime. Cities such as Oakland, California, have seen immigration nearly double while violent crime has fallen by more than half. The evidence here and in other studies pretty clearly contradicts the notion that immigration increases crime. In fact, other studies have found lower crime rates in high-immigrant areas and in so-called "sanctuary cities," and that immigrant populations on the whole tend to commit crimes at rates somewhat lower than the native-born population. (See this previous post on evidence that sanctuary cities tend to have lower crime and better economies.)
This of course doesn't mean that there are no criminal immigrants, or that the drop in crime is attributable mainly or entirely to immigration. Nationwide, including in areas with relatively few immigrants, crime has fallen a great deal since roughly 1990 for reasons that aren't clear and likely have to do with a combination of things. (See this post from last fall.) Kevin Drum has argued fairly convincingly that one major factor has been the reduction in lead in the air resulting from the switch to unleaded gasoline. Lead compounds can damage children's brains in ways that decrease impulse control, which in turns increases tendencies toward violent crime. For a summary of his evidence see this link.
Below is a 4-minute report from Sunday's edition of PBS NewsHour about the Marshall Project analysis.
(Incidentally, ignore the dumb thumbnail picture which, unless they've changed it, makes it look like an anti-Trump piece. It isn't. It's a straightforward report that mentions President Trump only in passing, because he has sometimes implied erroneously that immigration increases crime.)