Given that a criminal is much worse off as an illegal immigrant (due to prejudice in any country against aliens, and the fact they can be deported even if found innocent), criminals generally prefer to follow their life of crime in their home country. By definition, a foreigner sticks out, and the last a criminal desires into stick out.
Trump shocked the Democrats by the respectable percentage of Hispanic votes he got. A Republican who sticks the core of Trump’s message, aimed for the working class (the “deplorables” in Democratic eyes), and avoids some of the off-color racist messaging that kept getting Trump in trouble, will undermine any Democratic advantage among new immigrants. It should also be noted that historically, it does not take very long for new immigrants to become nativists (and to start objecting to ever newer immigrants). Note by the way, that with a 19th century standard, Trump would be considered a new immigrant (1st and 2nd generation native to the USA, depending on parent).