They absolutely are pushing down wages, especially when they come from poor desperate countries, because they're in a much more precarious financial position, they're easier to exploit, less likely to join a union, and most importantly, more likely to work for lower wages.
Separating that, from the notion of racism or people getting mad at the immigrants, and not the elected officials creating the policies that allow your boss to import and exploit people from overseas, is important to working class success.
How did we get so easily baited into falling for this?