Do you have any current data on that? I think it was true of the "Reagan Democrat" variety of union worker in the 1980s, but I suspect it's an outdated stereotype now. For example, SEIU, one of America's largest unions, represents a lot of immigrants and is strongly pro-immigrant: https://www.seiu.org/justice-for-immigrants/
It's not really true any longer, but it was at one point. The head of the AFL-CIO admits here that unions were strongly anti immigration for decades. Only in the past few years have they changed their official opinion.
There's a big difference between being anti-immigration and anti-immigrant. All of the latter people are also the former, of course. But there are people who want lower immigration even though they have no problem with the immigrants who are here. Some union people are surely in that bucket just for "reduce job competition" reasons.
The only reason blue collar workers' unions changed their tune is because they were unable to stop the flow of illegal immigration. On the other hand, restricting H-1Bs and green cards is a battle that many think they can win.
