The article presents a graph where two metrics are overlapped on the same scale (union membership and percentage-based middle-class income), with a graph title suggesting that one causes the other. This isn't necessarily true.
Irrespective whether you agree or not with the article's arguments, in general correlation doesn't imply causality.
The drop in the percentage of middle class income could be caused by a bunch of factors, including (but not limited to) increased automation capabilities and increased reliance in off-shore low-wage workers.