He'd easily get the nomination, but like others have said I don't think he'll beat Bush, although he'll probably do better than Kerry did (he may win Iowa and New Mexico like he did in 2000, Kerry lost both of them in '04 OTL.)
If for whatever reason he does win ITTL 2004, he'll be seen as the second coming of Jimmy Carter because he'll get the blame for everything that went wrong from '05-'08 even though it wouldn't be his fault. He'll get blamed for Iraq, the economy, gas prices, and Katrina (the state and local govts responded to Katrina as badly as Bush did in OTL, I don't see this being any different ITTL), and as a result of this, 2008 ITTL will be a repeat of 1980. The incumbent Democrat will get crushed by the Republicans.
This is why in hindsight (and I say this as a staunch Democrat), I'm glad Bush won in 2004. He rightfully (and in some ways wrongfully) got the blame for the sh*t that happened from 2005-8 and as a result, the GOP and the conservative movement was crippled, and the Democrats won big in 2008.