With few exceptions, immigrants are less crime prone than natives or have no effect on crime rates.

See Immigration and Crime – What the Research Says by Alex Nowrasteh of Cato. Excerpts:

"With few exceptions, immigrants are less crime prone than natives or have no effect on crime rates.  As described below, the research is fairly one-sided."  

"Butcher and Piehl examine the incarceration rates for men aged 18-40 in the 1980, 1990, and 2000 Censuses.  In each year immigrants are less likely to be incarcerated than natives with the gap widening each decade.  By 2000, immigrants have incarceration rates that are one-fifth those of the native-born."

"Between years 2000 and 2005, California cities with large inflows of recent immigrants tended have lower violent crimes rates and the findings are statistically significant.  During the same time period, there is no statistically significant relationship between immigration and property crime."  

"Ewing, Martinez, and Rumbaut summarize their findings on criminality and immigration thusly:

“[R]oughly 1.6 percent of immigrant males 18-39 are incarcerated, compared to 3.3 percent of the native-born.  The disparity in incarceration rates has existed for decades, as evidenced by data from the 1980, 1990, and 2000 decennial census.  In each of those years, the incarceration rates of the native-born were anywhere from two to five times higher than that of immigrants.”

"In 2010, 10.7 percent of native-born men aged 18-39 without a high school degree were incarcerated compared to 2.8 percent of Mexican immigrants and 1.7 percent of Guatemalan and Salvadoran immigrants.  These are similar to Rumbaut’s older research also based on Census data from 2000.  Controlling for relevant observable factors, young uneducated immigrant men from Mexico, El Salvador, and Guatemala are less likely to be incarcerated than similarly situated native-born men."

"The phased rollout of the Secure Communities (S-COMM) immigration enforcement program provided a natural experiment.  A recent paper by Thomas J. Miles and Adam B. Cox used the phased rollout to see how S-COMM affected crime rates per county.  If immigrants were disproportionately criminal, then S-COMM would decrease the crime rates.  They found that S-COMM “led to no meaningful reduction in the FBI index crime rate” including violent crimes." 
       
"“[v]iolent crime is not a deleterious consequence of increased immigration.”  Martinez looked at 111 U.S. cities with at least 5,000 Hispanics and found no statistically significant findings.  Reid et al. looked at a sample of 150 Metropolitan Statistical Areas (MSAs) and found that levels of recent immigration had a statistically significant negative effect on homicide rates but no effect on property crime rates." 

"Wadsworth found that cities with greater growth in immigrant or new immigrant populations between 1990 and 2000 tended to have steeper decreases in homicide and robbery rates."

"violent crime rates tended to decrease as the concentration of immigrants increased.  An immigrant concentration two standard deviations above the mean translates into 40.5 fewer violent crimes per 100,000 compared to a decrease of 8.1 violent crimes in areas that experienced a change in immigration concentration two standard deviations below the mean.

"Mexican rainfall patterns and the subsequent immigration had no effect on violent or property crime rates in major U.S. metropolitan areas.  

"Davies and Fagan looked at crime and immigration patterns at the neighborhood level in New York City.  They find that crime rates are not higher in areas with more immigrants.  Sampson looked at Chicago and found that Hispanic immigrants were far less likely to commit a violent criminal act then either black or white native Chicagoans.  Lee et al. found that trends in recent immigration are either not correlated with homicides or are negatively correlated in Miami, San Diego, and El Paso."

"Numerous studies also conclude that the high immigration rate of the 1990s significantly contributed to the precipitous crime decline of that decade.  According to this theory, immigrants are less crime prone and have positive spillover effects like aiding in community redevelopment, rebuilding of local civil society in formerly decaying urban cores, and contributing to greater economic prosperity through pushing natives up the skills spectrum through complementary task specialization."