The debate on the debt ceiling could take one of two courses:
Option 1: The White House and congressional Democrats take a firm position that a vote on the debt ceiling must be "clean," i.e. not with any attached provisions on budget cuts Republicans want. With the business community begging Republicans not to undertake a possibly unsuccessful blackmail strategy that could involve the United States of America defaulting on its debt, therefore initiating a global financial crisis, the Republicans are backed into a corner, as it becomes clear that they are willing to put their highly ideological agenda before the country's economic well-being. As the countdown to the date approaches, the alternatives become clearer and clearer, and eventually they have no choice but to allow the debt ceiling increase to pass.
Option 2: The White House and congressional Democrats take the position that they'd like a clean debt-ceiling vote, but they're open to negotiations about attaching budget changes to it. Republicans make demands, then escalate those demands. In the end, the two sides reach a deal, one that contains painful cuts to important programs and allows Republicans to claim victory, since they advanced their longtime goal of squeezing government. Once again, the Democrats have been rolled; in an attempt at damage control, President Obama comes out and says, "The American people are the real winners here, since we averted disaster."
Option 1 is a political and substantive victory for Democrats. Option 2 is a political and substantive victory for Republicans. Which one ends up happening is completely in Democrats' hands; if they simply hold firm and refuse to negotiate over the budget until the debt ceiling vote is out of the way, we won't be talking about what kind of budget alterations we ought to make, but rather, we'll be talking about the consequences of the Republicans' dangerous blackmail. And which one are Democrats going to choose? Why, Option 2, of course.