It is not inconceivable that he would have been nominated over Kerry, provided he started early enough and had the organization. While he is a Democrat with a conservative record, to be elected he would somehow have to prove he could be a better commander-in-chief than Bush and handle the Iraq War and Terrorism. While he couldn't be "swift boated," he would have to stand-up to the Rove destructer machine.
They might try to make something out of the fact that he was a corporate lawyer during the period between his governorship and his election to the Senate. That's always a way to make him look a villian to the average voter, especially if they could tie the firm to something questionable or unsavory, even if Bayh himself was not involved with it. [i.e. in 1997 Baker and Daniels defended Dr. Smith who advocated assisted suicide on demand. Evan Bayh chose to join this firm and made over $200,000 while working there, lobbying for corporations and abortionists. What does that say about Evan Bayh's values?"]
Simply winning Indiana (11 EV) (which IOTL Bush took by 500,000 votes) wouldn't be enough, it would still come down to Ohio.