As Matt's piece implies, holding the more popular position -- as the Democrats do -- is a necessary, but not sufficient condition for winning the sequestration fight. See Paul Begala for more.
But there's much more at play here than public opinion, the relative popularity of a "balanced approach" to deficit reduction, and the challenging environment some Democrats will face in 2014.
The most important factor in this fight is probably the reality that Obama doesn't have to face voters again and thus is willing to veto sequestration replacement bills if they're composed of spending cuts alone. Congressional Democrats are fully aware of this, too, and that creates a powerful incentive for them to hold the line.
So sequestration will begin. Obama won't cave. And then the tension sequestration was intended to create -- and in fact has created -- between defense hawks and the rest of the GOP will intensify and actually splinter the party. If that doesn't happen quickly enough, then the sequestration fight will become tangled up in the need to renew funding for the federal government at the end of March. If Republicans don't cave before then, they'll precipitate a 1995-style government shutdown, public opinion will actually begin to control the outcome, and it'll be game over.
So there are real dynamics at work here that can break the GOP's resolve in this fight but that can't easily be turned against Obama. Which means even though months of sequestration and a government shutdown followed by Obama folding outright is a theoretically possible outcome, there's very little about the nature of the fight to make me think it's likely to happen.