Answer:
By showing them that they're serious about the games.
Explanation:
In the book, Katniss and Peeta do a couple things to get Haymitch to take them seriously. The initial confrontation on the train (where there's a verbal argument and Haymitch punches Peeta before Katniss drives her knife into the table near Haymitch's hand) lets Haymitch know that both tributes have a fighting spirit. After that, both tributes don't take Haymitch seriously until he starts to take them seriously in return.
