"All of us assign blame in our own best interest—blame is relative. So one of the most important functions in society is controlling the blame pattern. Why is it that [the working class] assign blame downward to some welfare chiselers down at the bottom, "Tryin' to get a little somethin' for nothin' and they never assign blame upward to a handful of big-time chiselers who get a whole lot of something for doing nothing at all?" - Utah Phillips, IWW activist and folk singer
Stop blaming the people who have no control over the system and start looking at people who really do. Blaming immigrant workers for low wages and lost strikes is playing the employers' game. In a previous era it would have been accusations against African American workers or the Italians, but now it is the Mexicans.
In 1986, Congress passed the Immigration Reform and Control Act (IRCA), which imposed penalties on employers for hiring undocumented workers. The idea was in fact supported by the AFL-CIO at the time although the legislation is essentially based nativist and racist logic. By making it a crime for employers to hire workers without papers, the workers would be unable to work and would leave the country. Immigration would taper off because people wouldn't come to the United States, knowing they wouldn't be able to work. Job competition would slack off, and wages would go up because "they" wouldn't be taking "our" jobs. Some unions (United Electrical, International Longshore and Warehouse Union, garment unions) disagreed with that AFL-CIO position. They predicted that sanctions would simply make workers more vulnerable and be used as a weapon against immigrant workers when they tried to organize. They argued that instead of uniting workers, it would divide them against each other. Their predictions were borne out. Sanctions were used to fire immigrant workers when they tried to organize. Sanctions had no effect on slowing migration. Nor did they end job competition. Instead, they gave employers an even greater advantage when the Supreme Court held that employers didn't have to rehire undocumented workers fired for union activity or pay them back wages.
In 1999 the AFL-CIO federation eventually passed a resolution calling for the repeal of employer sanctions and for the legalization of people in the United States without papers. Nevertheless, Democrats have proposed immigration reform bills (and some unions have supported them) that would strengthen employer sanctions. We already have an electronic verification program, E-Verify, a database maintained by immigration authorities so that employers can check the immigration status of workers.