It isn't just republicans who want cheap, democrats also own businesses to and wish to exploit that dirt cheap labor.. Many states also want illegal immigration because it inflates their population count. Which means more representation with a smaller voting pool when its census time. This is why as citizenship question is a great idea for the census so that only citizens can be used to apportion representation and tax dollars.
I wouldn't say dirt cheap. In fact many times they get paid quite well. Often they get paid piece work. Their production is 3 or 4 times that of American workers. That production rate pays off quite handsomely.