Republicans—shying away from the consequences of the government shutdown they provoked—have started rallying around the idea of addressing specific problems with a government shutdown by passing mini-appropriations to rescue particular agencies. Helping out veterans and reopening national parks, in particular, seem to be on the agenda.
Democrats, meanwhile, are resisting this approach for a mix of a few reasons. One is that this is basically a silly way to run the government. The other is that Democrats think they're winning the political argument right now and don't want to change course. And the third is that they fear these small-batch appropriations might be a viable strategy for the GOP: reopen the parts of the government Republicans don't mind, while letting vital programs like WIC or agencies the GOP hates like the EPA wither on the vine.
This last is, I think, a mistake.
The cliché about public opinion and government spending is that the American people are "ideologically conservative and operationally liberal." That means Americans absolutely agree that government spending is much too high, they just don't happen to favor spending cuts on any particular program. This Pew survey from February was typical. Even the dread foreign aid doesn't quite get majority support for cuts. And support for increasing spending exceeds support for cutting spending in virtually every category. That includes food inspections, environmental protections, aid to the American poor, and basically every other key category.
Veterans' benefits happen to be the single most broadly popular category of spending, so it's no surprise that would come first on a piecemeal list. But going line by line through the government puts Republicans on precisely their weakest ground—specifying where, exactly, they want to cut in a country where the public doesn't want to cut anything in particular.