Many of the immigrants, both legal and illegal are escaping to America to get away from the crime in their home country, so they are not here to cause any trouble. The only even remotely immigrant related increase in crime is when the gangs immigrate here to recruit or to tighten their grip on the surrounding countries.
Immigration alone does not lead to increased crime because we have a huge list of steps that need to be taken for someone to be allowed into our country as a citizen. Illegal immigration might lead to increased crime rates because we can not filter our the drug dealers, and smugglers, but immigration itself does not.
I believe that the average immigrant is more likely to try to stay out of trouble, because they are aware that breaking laws here could mean deportation. I believe that immigrants are more aware of the laws in the country that they move to than the average citizen is, simply because the consequences of breaking those laws could be so much more severe.