In melee, among the top 15 players we see Fox, Falco, Puff, Peach, Sheik, Marth, Pikachu, Yoshi, Falcon, IC's.
Even in the top 6 players we see them using 6 different characters (the first 6 in the above list).
Among those, the better player wins, or the one playing better at the time. Fox has an edge in some of those MU's, but that rarely stops the better player from winning. The top 5 (right now 6) players have been so consistent with these characters over the past 5 years, each beating the other top players frequently enough, that it indeed is a very balanced game.
If, however, by balance, you mean that you can use any character you want, even characters intended to be bad, the game is not balanced. But only newer players seem make that argument. If we are talking about the usable characters, and there are at least 6 of them that can consistently win a major tournament, the game is very balanced.
Chances are at your skill level all the games are balanced. Back home I would play with my buddies and we would basically stay within the characters of: falcon, ylink, luigi, falco, pikachu, doc, fox, marth, and ganon, and they were all even to us. Falco might have been slightly better due drill shines and crouch cancel to fmash(at least in our group at the time), but that rarely ever stopped the one who played better from winning.
So chances are at your level, and if you played for a long time practicing, melee would remain pretty balanced among most characters if you played with people around your same skill. As you got better, you'd notice the game remains pretty balanced among the top 6-10 characters, with the top 6 specifically among the very best players.