The problem isn't that joblessness is stemming from illegal aliens taking away jobs (much less jobs that pay a livable wage and possess any sort of benefits), it's that such jobs are being offshored to other countries.
If there is a problem with Mexicans doing manufacturing jobs that Americans used to do, it isn't because Mexicans crossed the border illegally - it's because companies shipped their factories to Mexico where they can pay people less and environmental and labor standards are less stringent. It's the fault of the companies, not the people.
But people who get wrapped up in xenophobia tend to ignore that, since they are more interested in rationalizing their racism than any actual concern about the well being of American workers.
(leading to a great political irony as it regards immigration - Democrats tend to be stricter about border enforcement, but play it down for fear of looking anti-immigrant, Republicans play up the rhetoric for the [thinly veiled bigotry] of a part the base, while looking the other way as many big donors can save lots of money by hiring illegally).