I think you should consider the issue of inequality and how relatively extreme the disparity is in wealth and income between the rich and the poor in the US. It's an ugly curve.
http://assets.motherjones.com/politics/2011/inequality-p25_averagehouseholdincom.png
Unions may not be a cost-free way to improve the slope of the curve, but we need ways to do so and they are one of the few tools we have that have an impact. For instance, where there are massive inequalities in compensation for CEOs and workers, unions can pressure chiefs over the inflation of their income - so long as the ratio is disclosed.
http://www.epi.org/publication/ceo-pay-continues-to-rise/
Why should you want that? Wealth concentration is highly likely to be economically inefficient. For the whole of a society. In fact, societies which concentrate wealth as a matter of history appear to create all sorts of problems, not the least of which is chronic instability. Think of the Russian and French revolutions and gross inequality is right there at the heart of the issue.
Another perspective: wealth isn't merely measured in the depth of your pocket. Societies share wealth in the form of roads, health systems, fire brigades, law courts and other public goods. It's hard to have a nice life in a country without laws, freedoms and community assets. Taxes aren't merely a cost. They are the source of shared benefits as well.
So it's about finding a balance. Neither socialism nor capitalism creates pure virtues. They both require modification to reduce the harms they generate.
If your economy concentrates wealth (which seems to be a natural process of unregulated markets), your production turns to luxury items like yachts and private jets for the wealthy, while workers eat fast food and the poor commit survival crime. Sound familiar?
If wealth is spread more broadly (and that doesn't mean the same thing as equally), there are more consumers with spare cash, there are more folks investing in small businesses, there are more kids pursuing studies etc. Fun, highly educated culture, strong economic growth. Unfortunately, that's more Scandinavia than the US.
http://people.duke.edu/~dandan/Papers/Other/BuildingBetterAmerica.pdf
One would rather not solve issues of gross inequality through redistribution, regulation or intransigence. But if owners fail to spread the wealth from productivity gains and we know trickle down doesn't work, then the options are limited.
Frankly, public unions are a minor issue in modern America. This ain't the 1970s.