The ability for the firemen to exist as they are in Montag's society, as men who burn books, was paved long before that job existed. Society paved the way for firemen to be possible. Beatty explains the road leading up to firemen when he visits Montag's house the morning Montag stays home sick (after they burn Mrs. Blake with her books). Beatty explains that people got tired of reading so much, of putting so much effort into reading, so books got shorter and shorter, and finally people just stopped writing them because no one bought them or read them. So that is one factor. Another is that so many people were offended by the various content of books, that they started banning them, censoring them, and sometimes even burning them themselves. They were offended by them, so reading them became the uncool thing to do; if you read those hate-mongering books, you were a racist, or insensitive, or prejudiced. Also, if you read books you were considered snobby because you made people who didn't read look and feel stupid. The stigma placed on you if you read books was so uncomfortable that people stopped reading to avoid being made fun of and ostracized. So, books became unpopular in that way. All of that paved the way for firemen.
Montag asks at one point, "in the old days, before homes were completely fireproofed--" whether firemen prevented fires instead of starting them. So, they bust out the rule book, which states that the first fireman was Benjamin Franklin who, in 1790, burned "English-influenced books in the Colonies." It's an interesting assertion. Then, the rules state:
"1. Answer the alarm swiftly. 2. Start the fire swiftly. 3. Burn everything. 4. Report back to the firehouse immediately. 5. Stand alert for other alarms."
So, pretty simple rules there, but getting to the point where firemen became what they were, required a long, slow process that was forged by society's apathy and political correctness, along with modern technology that made homes "fireproof". I hope those thoughts help! Good luck!