Okay, let's put this into few categories (also note that this is just my opinion):
Character design and personality
Brock - Shirtless and with his eyes closed, he is the 1st Gym Leader that most of us encountered in the games. He's considered as one of the strongest trainers around Pewter City. I don't really like his early design in the games and his HGSS and B2W2 design is okay at best. He seems really ordinary to me and nothing screams "THIS DUDE IS MORE IMPORTANT THAT THE YOUNGSTER YOU JUST BEAT!".
Falkner - Now we are talking. Falkner's design really fits the whole atmosphere if the Johto region and Violet City. And it's even consistent throughout the games.
FALKNER WINS THIS ONE
Pokémon
Let's talk about Pokémon team when you face them:
Brock - He uses Geodude and Onix. None of them know any Rock-Type attack in RBY and only Onix knows Rock Tomb in the remakes. His team is on levels 12 and 14 and only 10 and 12 in Yellow. These levels and Pokémon's Types give nice challenge to everyone, because they can't just randomly choose attacks - there is small chance that they will be super effective, but they mostly won't. You'll also see Pokémon you haven't seen before during your short journey, which is nice bonus.
Falkner - He uses Pidgey and Pidgeotto that is underleveled for some reason. In the Gen II games, he is big disappointment. Your Pokémon will certainly be at least at level 10 and his Normal and Flying-Types attacks won't do much. In HGSS, the situation is a little bit better with his Pidgeotto being on level 13 and knowing Roost. But still, you can pretty much solo him with just Chikorita without paying much of the attention on what is going on. And he uses probably the worst Flying-Type he could - He could have any of the new Flying-Types from Johto and it would make the fight more interesting.
BROCK WINS THIS ONE
Gym design and grunts
This part will be short. Both of them are boring. I get that they're there to introduce players to the whole concept of Gyms, but wouldn't it be better to include small puzzle in there? But I'll have to give this one to Brock, because of the fact that once again, his grunt shows of new Pokémon to the player. That is something that even the whole upper level in the HGSS won't beat.
BROCK WINS THIS ONE
Actual battle
Now, let's look at the battle using the teams above (Pidgeot sprite is only one that is animated btw and it drives me crazy). Falkner has literally only one Pokémon that can counter only half of Brock's team. Going with the levels, Brock's Pokémon are around level 55 (with his ace Onix on level 61) and Falkner's Pokémon are around level 52 (ace Pidgeot on 56). Falkner's Staraptor and Swellow could do some damage, but they would have to 1HKO Brock's Kabutops, Omastar and Relicanth to make room for Pelliper to finish of the rest of the team, but I can't see that ever happening. Maybe Falkner's Honchkrow could land few hits with Sucker Punch, but that would hardly make a difference, because Honchkrow is on the same level as the Brock's weakest Pokémon.
BROCK WINS THIS ONE
And so that's my analysis. I personally think that Brock would not only win in the battle against Falkner, but introduces players better to the whole world of Pokémon. Even though he has terrible design.