A Dystopian genre is defined as an upside down world, controlled by some form of government & has some crisis going on in the world. The way The Hunger Games are considered a Dystopian genre style. tTilogy is because of multiple factors such as: Government. The Capitol is the controlling force in the trilogy, and they use their remarkable arms forces to control all 13 Districts (12 after the dark days). The crisis being The Hunger Games itself, which is basically an annual fight to the death, where two tributes, one boy and one girl (aged 12 to 18) get called from every district in a lottery style event and are entered into the fight to the death. This is also known as the annual reaping.
The setting Panem is considered a rather totalitarian nation and is divided into 12 Districts, and all of the Districts have a key role in the functionality of the nation. I.e. District 12 is the mining aspect of the nation, 11 being agriculture and so forth. The whole system works granted, but is also considered nevertheless Dystopian in itself, we see this because if someone refuses to work, they are punished, treated as commodities shot dead showing anyone else in the District that if they rebel they pay with their life.
Katniss (The main character in the trilogy) is an unlikely hero herself as when the annual reaping came around, and Katniss’s sister Prim had been pulled as the girl representing her district Katniss knew straight then on the spot, what she had to do. She sacrificed herself and volunteered as tribute to save her sister. Moreover, leading to herself being entered into The Hunger Games. Also In the other movies, Katniss goes on to rebel more against The Capitol, and is the key person who unites all The Districts and pursues them to rebel against The Capitol. Without her, none of this would be possible as her confidence, and recent scenarios of her rebellion against The Capitol added to The Districts trusting her and wanting to help. Like when she invaded the rule change to have two victors in the end of the first movie, these are just some key attributes that show that Katniss is an unlikely hero.
Control; the control in the trilogy is defined as The Capitol, which is a government of sorts, the way these people are controlling The Districts are the guards, formally known as Peacekeepers. These people are heavily armed with weapons and armour. They force the people in the Districts to work for money, food, etc. Not to mention the reaping where they force two people from every District, one female and one male aged 12 to 18 to be entered into the games. It is not optional. When in the games if they do something that is disliked by the Game makers (people controlling the games), they will do this such as throw fireballs as was seen when Katniss was approaching the border of the arena and the Game makers wanted her to go the other way etc. Also the citizens are restricted to just their respected District, and if anyone is caught attempting to sneak out of the District they will be punished with their tongue cut out, leading them to be turned into Avoxes or worse, killed.
This is just some of the reasons that further the point that The Hunger Games are a Dystopian genre style trilogy, it has clear forms of control, there is a crisis (The Capitol) and an unlikely hero rebelling against them (Katniss Everdeen) and more.