Eh, this argument gets trotted out a lot, but it isn't exactly clear-cut. The economy isn't a zero-sum game.
Sure, illegal immigrants do show up and undercut some jobs. Some percentage of people will find themselves working at a lower wage, or lose their job and need to find a new one.
But plenty of other people will find their business increased, higher wages, more profit, etc. All these immigrants require services and commodities too, and have to buy them from somewhere. They need to buy groceries, get hair-cuts, purchase mattresses, buy cars. Illegal or not, they contribute to the economy too.
Planet Money did a good piece on the cuban migrations to Miami in the 80's. It was the perfect microcosm to study the phenomenon: many immigrants, contained geographical region, etc.
http://www.npr.org/2015/10/01/444912593/when-cuban-migrants-...
Money quote at the top of the episode:
SMITH: So 80,000 refugees came to Miami, and there was no effect on the economy?
CARD: You couldn't detect any strong evidence of an effect one way or the other.
SMITH: No effect, not on wages, not on unemployment rates. Card says this is
because there is not a fixed number of jobs in an economy. When a bunch of new
people come, yes, they get jobs. But they also start buying stuff, getting haircuts,
going to the grocery store, which means you need more barbers and more grocers. That
creates more jobs.