I wouldn't have thought that the question of why working class marriage rates have fallen would be a sharply partisan issues. But since Charles Murray decided that moral decay is to blame, many of his critics have started countering that poor economic outlook for working class men over the past decade is the issue. The obvious third hypothesis, it seems to me, is that with women's social and economic opportunities much improved over the past several decades they're less reliant on marriage as a form of social and economic support.
Some backing for that thesis can be found in Scott Hankins and Mark Hoekstra "The Effect of Random Income Shocks on Marriage and Divorce" which studied lottery winners in Florida:
Results indicate that large income shocks significantly reduce the likelihood that single women marry. Specifically, we find that single women are six percentage points less likely to marry in the three years following the positive income shock, which represents a 40 percent decline. This suggests that additional income may remove some incentive for single women to marry, at least over the short-term.
In contrast, large cash transfers do not affect the marriage rates of men or induce couples to divorce. We find that the divorce rate of married recipients of $25,000 to $50,000 in the three years following the income shock is between one-half and one percentage point lower that that of recipients of $1,000, which is small both in absolute terms and relative to the baseline 3-year divorce rate of 8.5 percent. Estimates also allow us to rule out large absolute positive net effects of income shocks on divorce rates: even the upper bound of the 95 percent confidence interval implies that fewer than 1 in 70 married couples will divorce due to the positive income shock of $25,000 to $50,000.
That's hardly the last word on this, but it seems to me like the correct place to be looking. Austen (1813) argued that "it is a truth universally acknowledged that a girl of a certain age, and in a certain situation in life, must be in want of a husband." If social and economic trends over the past several decades have reduced the supply of girls in that "certain situation in life" then marriage rates should decline completely regardless of what happens with men.