The Democratic Party is in a much better shape to resist his message. People of Color make up a huge chunk of the Democratic primary vote, and are unlikely to be favorable to Trump; e.g. Hispanics are not going to go in large numbers for someone who describes Mexican immigrants as "rapists." It's very difficult to win the Democratic nomination without at least some support among those groups.
In addition, 2004 was heavily shaped by the Iraq War and national security issues, which are Trump's weakest point; polls suggest even his supporters don't really trust Trump with nukes, and that will be far more in people's mind in 2004 (before the Great Recession, and with violence in Iraq regularly making the news) than it is now.
Even if Trump somehow did as well ITTL as he did for the Republicans this year, the Democrats also have super delegates (and this is pre-2008, so they actually have a ton of super delegates, more than this year), who could and would be used to deny him the nomination. This isn't a situation like 2008 or 2016 where the delegate leader was a candidate that the establishment could accept (even Bernie, if he had won the most delegates, would have been tolerable for the DNC, even if he wasn't their first choice); this is a situation where the super delegates would come down pretty decisively against what they see as a racist, sexist demagogue.
But the biggest problem is that The Apprentice didn't launch until January 2004; Trump hasn't really been visible enough to command instant popularity, and to the extent he has name recognition, much of it will be for his 1990s bankruptcies (the last time he was in the news in a big way), rather than as business guru and host of the popular reality TV show.
So essentially, he never gets anywhere. Now, Trump in 2012 (when he also toyed with running as a Republican) might be interesting.