Hard to say since amongst the Crusaders were the Normans who had fought against Alexios and would probably still try to portray the empire in a negative light. Sure if the emperor makes it to Antioch and before returning to Constantinople provides the Crusaders with the logistical support to proceed in their offensive, then it would be much harder for them to spread the word of the “treacherous deeds of the Romans” amongst Western Europeans. After all Alexios never committed himself to march to Jerusalem alongside the Latins. Probably the best that could happen however, would be for the Crusade to fail achieving anything after Antioch and the survivors be forced to join the empire and replenish the ranks of its armies, basically what Alexios really wanted from the Crusade.