Warning...this is not rigorous. This is just me playing with data sets that were easy to google because I was curious and couldn't sleep. There are probably legit papers that provide much more accurate answers.

First, let's check the title question. What do we get if we plot crime rate vs foreign-born population? Defining crime here as total number of murders, rapes, robberies, and aggravated assaults per 100,000 people, you get the following:

There's a slight negative slope.  If you aren't familiar, the way you'd read that is that knowing nothing else about two cities, if one has a 10 point higher immigrant population percentage, you'd expect to see 124 fewer violent crimes per 100,000 people in that city every year. The R^2 is only 0.12 so that's pretty low and implies that the relationship isn't very strong/there are other, major factors.

An obvious question is how this differs if you consider only illegal immigrants. I couldn't find as much data for that, so here's the same plot for 61 cities considering only illegal immigrant levels:

For completeness, here's the original plot for foreign-born population with just those 61 cities:

Same basic trend.

There are a lot of other factors that came to mind, so I decided to put them together and see if any factors were well-correlated with violent crime. I used total population, foreign-born population, illegal immigrant population, GINI index, and black population stats for 34 cities. This is not very many and this is not rigorous analysis at all, but it's fun to run the numbers. 

Based on that, the overall regression r^2 is 0.43. The only stat with a p value < 0.01 is 'black population %', and the coefficient for that stat is 18. The confidence interval for it is 8 to 27. Like before, the crude interpretation is that if you know nothing else about two cities, if one has a 10 point higher black population percentage, you'd expect to see between 80 and 270 more violent crimes per 100,000 people in that city every year.

Foreign-born population was the only other one that came close but the p value is 0.08 (coefficient was -18).

I used the following data sources. Note that when a city source was unavailable, I used the stat for that metro area so those are even less accurate.

• foreign-born: https://en.wikipedia.org/wiki/List_of_United_States_cities_by_foreign-born_population
• illegal: https://www.pewhispanic.org/interactives/unauthorized-immigrants-by-metro-area-table/
• gini: https://www.brookings.edu/research/city-and-metropolitan-income-inequality-data-reveal-ups-and-downs-through-2016/
• black: https://en.wikipedia.org/wiki/List_of_U.S._cities_with_large_African-American_populations
• crime rate: https://en.wikipedia.org/wiki/List_of_United_States_cities_by_crime_rate

• very limited data set for black populations
• had to mix metro and city data
• data is not all from the exact same time periods
• few cities overall...I'd ideally have a couple hundred US cities
• lots of factors I didn't find data sets for in the limited time I looked...e.g., 'average age', 'average air quality', and 'average July heat index' came to mind as possible contributors