Doug, quoting Bartels:
"White voters in the bottom third of the income distribution have actually become more reliably Democratic in presidential elections over the past half-century"
Does 'the bottom third' correspond to the 'working class'? It rather depends on whether you are using a Galbraithian, two-thirds/one-third model or a Marxist non-owner of the means of production model. The Marxist model would see the working class as rather larger than one third of society, I think.
On the UK model. I would suggest that the working class is less organised, and less organisationally committed to the democrats, its vote flabbier and more disengaged, but at the ballot box, probably more likely to vote democrat. I don't know this to be the case, but my guess is that the analogy with left identification in Europe holds good.
That, they do vote democrat, though, is a register of their ideological weakness, not their strength.