I don't remember democrats representing the working class in any meaningful sense.
They did, though ... and more to the point, they created the impression that they were the party of the common man.
Democrat support for and from unions has its roots in the early days of the labor movement, at a time when unions actually did some good. The GOP was seen -- properly, for the most part -- as having opposed it. People in the 60s and 70s had grandparents and parents who lived through that time.
As another example, I don't think many of us on this board have any idea of what true, grinding poverty looks like -- before, during, and after the Depression. But we know people who did. The Democrats attempted to address it, and the GOP opposed it. People now alive know, or even lived through, that period.
Of course the Democrats also screwed things up, badly ... but we're talking about impressions here, and perceptions tend to lag well behind the long-term consequences.
Reich's point, even if he doesn't realize it, is that today's Democrats look an awful lot like the GOP of old -- playing to the wants and needs of their rich pals, and scorning the lower-class "rubes" who haven't the benefits of their Yale women's studies degrees, or high-class sensibilities.