Consider some basic logic:
1. I'm agnostic on what the exact employment effects of a higher minimum wage would be, but I do know this: A modest increase in the minimum wage probably won't mess up the labor market too much, but it also won't have much impact on poverty. On the other hand, a large increase would make more of difference in the lives of full-time minimum wage workers--but it's also far more likely to lead to a significant reduction in the number of jobs for those workers. This unavoidable tradeoff makes the minimum wage an inherently bad tool for making low-skill workers better off. Economists have long preferred the Earned Income Tax Credit, because it supplements the wages of low-skill workers using the tax system.
2. The most important labor market phenomenon of the last decade or so is that it has become much easier to replace American workers with machines or foreign workers. So if I had to bet, I would predict that a higher minimum wage would merely speed up this phenomenon. It's just a bad idea in the long run to make employers pay more for workers than the market says they are worth.
3. Democrats presumably aren't championing an increase in the Earned Income Tax Credit (EITC) because it is perceived (rightfully) as more government spending. But the EITC is a better approach since there is no artificial deterrent to hiring low-skill wokers; employers pay the market wage for those workers, and we use the tax system to "top off" their wages, so there is not the same distortion in the labor market as created by minimum wage. The cowardice of the Democrats here is that they've opted for the minimum wage because it is perceived as "costless" when in fact it is anything but.
4. The only long term solution here is to make low-skill workers more valuable in the labor market. That's hard and it's expensive. It requires early childhood interventions, more aggressive school reform, more intensive job training for displaced workers, and the like. But rather than advocate this more sophisticated approach, the Democrats have offered something simple, and likely ineffective. But what's the point of caring about the fate of low-wage workers if the solution the Party decides to push is so superficial?
5. Worst of all, the Democrats tend to be on the wrong side of school reform. If American workers are to have a better future, then we need to reform education, particularly in urban areas. The teachers unions have stood firmly in the path of every meaningful school reform, and the Democrats have typically pandered to those unions. The Obama Administration deserves a lot of credit for Race to the Top, which has set in motion many meaningful reforms at the state level. But for the most part, the Democrats are still way too cozy with the major teachers unions, and those unions are way too obstructionist when it comes to making American schools better.
All of this is a long way of saying that the Democrats care about the right stuff here; any American who puts in a 40-hour week should not live below the poverty line. And yes, rising inequality will eat away at the fabric of American society.
But don't make the health care mistake in which a sensible objective (coverage for all Americans) has been totally botched. Instead, sell the American public on real policy solutions. Stop pretending the problem is simpler than it is.