First, Prince Prospero is terribly selfish, a trait certainly not associated with the heroic. Despite the fact that his "dominions were half depopulated" as a result of the terrible disease, Prospero does nothing to help his people. Although we cannot know what might be possible in terms of assistance, we see that the prince does not even try to stop the spread of pestilence. Instead, "he summoned to his presence a thousand hale and light-hearted friends from among the knights and dames of his court, and with these retired to the deep seclusion of one of his castellated abbeys." In other words, he gathered to him those individuals who were healthy and fun, friends whose company he could enjoy, and he left. He saved his own skin, caring nothing for his decimated kingdom and vulnerable subjects.
Second, Prospero is cowardly. Rather than bravely stay and take his chances with his people -- after all, one might say that a good captain goes down with his ship -- he runs. He runs far and he runs fast in order to save his own skin and to make that saving as pleasant as possible by throwing fantastic parties for his coterie of lucky friends.
Third, the prince is proud. He thinks that he can escape death, that his money and resources will protect him from the same fate that befalls everyone else. He seems to think that he is above his subjects rather than subject to the same caprices of mortality. When he sees the masked figure at his party, "his brow reddened with rage" as he asked who "dared" to insult him. His pride is immense, and it is hardly heroic.
Prince Prospero is a good example of the anti-hero because he is so careless with his life and the lives of his guests. He believes that the Red Death will not reach him because he is rich and sealed away in his castle. He does not stop to think that no one escapes the clutches of death. He doesn't even notice with the Red Death arrives as a guest at his party. His cavalier attitude about it ends up being what kills him and every single merry guest at his party. The typical hero would have found a way to save the day, but Prince Prospero is the exact opposite.
Note: an anti-hero is a protagonist who lacks the traditional heroic qualities and instead possesses qualities that are antithetical in nature to heroism.