I was wondering as to why only kids from each district are chosen to enter the Hunger Games instead of any of the adults?
From the first book:
(...) The Treaty of Treason gave us the new laws to guarantee peace and, as our yearly reminder that the Dark Days must never be repeated, it gave us the Hunger Games.
The rules of the Hunger Games are simple. In punishment for the uprising, each of the twelve districts must provide one girl and one boy, called tributes, to participate. The twentyfour tributes will be imprisoned in a vast outdoor arena that could hold anything from a burning desert to a frozen wasteland. Over a period of several weeks, the competitors must fight to the death. The last tribute standing wins.
Taking the kids from our districts, forcing them to kill one another while we watch—this is the Capitol’s way of reminding us how totally we are at their mercy. How little chance we would stand of surviving another rebellion. Whatever words they use, the real message is clear. "Look how we take your children and sacrifice them and there's nothing you can do. If you lift a finger, we will destroy every last one of you. Just as we did in District Thirteen."
In short: Taking children is a more frightening punishment for the defeated districts. From the interview with Suzanne Collins at: http://www.thehungergames.co.uk/about_the_author
A significant influence would have to be the Greek myth of Theseus and the Minotaur. The myth tells how in punishment for past deeds, Athens periodically had to send seven youths and seven maidens to Crete, where they were thrown in the Labyrinth and devoured by the monstrous Minotaur.
Even as a kid, I could appreciate how ruthless this was. Crete was sending a very clear message: “Mess with us and we’ll do something worse than kill you. We’ll kill your children.” And the thing is, it was allowed; the parents sat by powerless to stop it. Theseus, who was the son of the king, volunteered to go. I guess in her own way, Katniss is a futuristic Theseus.
I'm pretty sure the exact reason as to why only children are chosen is given in the book. I will research that and add it to this answer, should I find it.
Until then, maybe this quote from the Hunger Games Wiki may suffice:
Every year since the rebellion, the Capitol forced 24 children into the arena (...) in order to both entertain the Capitol citizens and remind the twelve districts how completely at the Capitol's mercy they are.
Like I said in my comment to the question, I think the fact that the tributes are children is just the icing on the "we can do whatever we want to you" cake, to make it that more evil.
As to the question in the comments, regarding volunteering, The Hunger Games Wiki has to say this:
By rule, once a person's name has been chosen to become a tribute, another eligible boy or girl may volunteer to take their place. (emphasis mine)
With "eligible" meaning "between the ages of 12 and 18" (taken from the same wiki page).
(Out of universe: The Hunger Games is a Young Adult trilogy. I can see how it would take away from that to let the teen-protagonist fight grown men and women. The way it is, it just fits the genre that much better.)
In the Epilogue of The Ballad of Songbirds and Snakes a reason for this is alluded to:
"It certainly supports her view of humanity," said Snow. "especially sing the children,,"
"And why is that?" asked Dean Highbottom.
"Because we credit them with innocence. And if even the most innocent among us turn to killers in the Hunger Games, what does that say? That our essential nature is violent," Snow explained.