Interesting question! In the 5-city California area I live in now, the immigrant populations--mostly Mexican, Indian, Pakistani, various Asian groups--are part of the solid bedrock of the citizenry, some with dedicated new housing developments of brand new modestly designed houses and apartment complexes and at least one small chapel-like church with spire on just about every block. I'm unaware of any increased crime.
The Maine town I've recently moved away from had no immigrants aside from the sprinkling of international students at the local 4-star college. The crime level was elevated by the descendants of long-long-term residents whose traditional sources of stable factory work have tried up over the last decades and left them basically stranded physically and economically (moving out of an area takes money; closed factories and the absence of adequate replacement work means no money).
The large and rapidly growing California town that was my hometown, had (when I was last there) several "transitional areas," as defined by pohnpei, that housed economically depressed immigrants, most of whom were invited by local churches' relocation sponsorship programs. There were precedent setting clashes between Western religion and Southeast Asian religious and shamanic medical practices. The poverty and crime in these areas were notorious and alarming to most of the immigrant families living there. These areas were in the older sections of town and also ill-kept and run down and dirty. In fact, this condition, and the blocks of shops with boarded-up store fronts, was a shock to those of us who had moved "north" and for one reason or another returned or drove through what once was pleasant tree-lined neighborhoods for solidly established citizenry.
Two new sectors of immigrant neighborhoods were burgeoning there around the time I went back East. The immigrants in one new sector, mostly Indian of either Christian or Hindu belief, were well established professionals who set up practices or businesses or elegant shops throughout town or in the area and bought top-of-the-line new housing, in the new sector, in elegant gated communities. The other, mostly university educated Mexican individuals and families, moved into transitional apartment complexes in well established areas surrounding the top two school districts. Crime was low in these areas as well as in the new sector mentioned previously.
Additionally, there was an ever-growing population of international students at the local California State University site and the handful or other local colleges. These were not involved in "transitional" style crime or poverty as most were from wealthy families and had little want--if any--while living in upscale housing communities near the university and colleges. [Hmmh--interesting survey. Go back to the 1950s, and I can tell you about the immigrant area I grew up in!!]