If they're coming in under a government program with proper paperwork, they aren't illegals.
Loose immigration policy and the lack of border enforcement obviously exerts downward pressure on wages for low skill workers. It also bids up rents since illegal immigrants are willing to pile into a 1 bedroom apartment. The elite own businesses and real estate, both of which benefit from illegal immigration reducing wages and increasing demand for rent. If you take a minute to think about the incentives, then see the effects in the world around you, it's pretty obvious what's going on.