Sort of.

There is no denying that the 'reds' and other left-leaning political orgs had a great deal of influence on the labor movement. There is also no denying that many of the early organizers in the industrial labor movement in the US had (at least) ties to some lefty org or another.

But I think you've got it a little backwards.

I think the balance of the evidence would show that the rise of third party ('red' or labor-based) orgs in the early 20th C brought about the move to vilify labor unions. Initial reaction to industrial unions was based on many things, but from a public-perception point of view, I'd argue that the main thrust of the argument focused on the anti-Americanness of, for example, the IWW.

There are a million nuanced disagreements out there with respect to the influence of reds in the labor movement. How those political actions and parties interacted with the average worker is the subject of enough stuff to fill a small library. As you might imagine, the arguments of each book tend to follow the political bent of the author. Folks who tend lefty see the early 20th C as a lost chance of sorts, one where the popular labor-political alliance was somehow quashed or suppressed by some person or organization in order to preserve the entrenched power structure. Folks who are a bit right in their views tend to see much of the early 20th C as a time of upheaval -- one where a European political impulse came to the US and failed. The reasons for failure vary, but most will argue that it's the result of popular disapproval.

There are many, many books out there that deal with this. Among the better: Salvatore, Eugene Debs, Citizen and Socialist, Sklar, The Corporate Reconstruction of American Capitalism, Halpern, Down on the Killing Floor, Storch, Red Chicago, and a million or so others.