Crime places a burden on our society, and crime committed by illegals particularly so given they shouldn't even be here in the first place.
From your own NPR link:
It is true that wages for low-wage workers have declined — they fell 5 percent from 1979 to 2013. That may not seem like a huge drop, but during that same period, the hourly wages of high-wage workers rose 41 percent, according to the Economic Policy Institute.
However, economists disagree over whether an influx of immigrant labor caused or contributed to declining blue-collar jobs and wages.
Borjas, Miller said, "went back and re-examined and opened up the old data, and talked about how it actually did reduce wages for workers who were living there at the time."
Borjas' new analysis found that the wages of high school dropouts in Miami dropped between 10-30 percent after the refugee influx (the analysis looked at 1977 to 1993).
But an earlier study on the boatlift, from 1990 by Princeton economist David Card, looked at wages of "less-skilled" workers overall (as opposed to just high school dropouts) and found "virtually no effect on the wages or unemployment rates of less-skilled workers, even among Cubans who had immigrated earlier."
The debate remains unsettled, and it's impossible to extrapolate the effect of the boatlift on Miami to the whole country.