I suspect RTW will be a boon to the most marginal of workers, such as grocery stores and similar. These guys pay minimum wage, no benefits, and then the worker pays union dues on top of that.Teenagers are the only ones ignorant enough to apply for such jobs. Part time work, taxes, and union dues combine to yield two weeks worth of work and literally a $10 net check. They usually quite after that non-sense.
As for other classes of workers (e.g. full time manufacturing), it may have little effect. I don't know for sure, I haven't looked at the unemployment data, which is always confused due to complicated dynamics. But typically introducing unions don't significantly change the net labor market prices. From that experience, weakening the unions probably doesn't change the net labor market price either.
In terms of politics though, its a big deal in cutting the democrats major funding source. I suspect that is what riles them up more anything else combined.