His family seemed to receive the raft of Snow due to his victor. Katniss seems to pull off the same stunt in Catching Fire, and the only reason her family aren't slaughtered in retribution is that the Capitol has bigger fish to fry, what with the revolution and all. Another quality of why the audience likes Haymitch so much is that he knows how to work the system and helps Katniss and Peeta.
The bond that develops between Haymitch and Katniss is rooted in their similarities. Being able to spot those similarities in Katniss is what inspires Haymitch to give a damn again, and to become the mentor she needs. Once the audience sees how similar Haymitch and Katniss are, they tend to become more like hearted toward Haymitch.
Haymitch goes through alcohol withdrawal so intense that Katniss is afraid that he might die. He went cold turkey from years of alcohol addiction in the name of a greater good. With Haymitch going through alcohol withdrawal to become a mentor for Katniss the audience emotions go out to him because he is better himself, something that the audience knows is a big step for anyone to live a better life. A lot of people may have experienced a scenario like that. A lot of the scenarios that Haymitch has gone through makes the audience love Haymitch Abernathy an antihero in the Hunger