If it's it put in those terms, Democrats have to lose. The public favors compromise, and if the Democrats hang tough and refuse to "compromise", then they will get the blame if the failure to reach agreement results in default and the resulting economic fallout.
The Democrats need to frame the issue correctly now. Raising the debt limit is not a partisan difference in priorities: the Congress has written a check for the budgeted revenues and spending, and raising the debt limit is just making sure the funds are in the bank to honor that check. For the Republicans to threaten to refuse to honor the country's commitments, and threaten the whole economy, is hostage taking. And any spending reductions they demand are a ransom to "free" the hostage (i.e. to preserve the creditworthiness of the USA and our present [fragile] economy).
It's not a question of partisan differences that can be resolved by "compromise". You don't compromise with hostage takers: you pay the ransom or you don't. (If you do negotiate, it's on the size of the ransom.)
So that's how the issue should be framed starting today: do the American people want to pay ransom to hostage takers, or don't they?
Update 1/4/13 Advice from a hostage negotiator.
Update 1/6/13 And in case you think the hostage/ransom metaphor is overly anti-Republican - note that the Republicans have used it themselves :
“I think some of our members may have thought the default issue was a hostage you might take a chance at shooting,” (Senator Mitchell) said. “Most of us didn’t think that. What we did learn is this — it’s a hostage that’s worth ransoming."