No, there is no good balance reason
The designers may have imagined there was; it’s conceivable they considered the option of Weapon Finesse as your 1st-level feat (or Weapon Focus, Exotic Weapon Proficiency, Quick Draw, and so on) to be important advantages of full-BAB classes. I tend to believe it was more of an “imagery” thing, though – full-BAB classes were the “weapon masters” and so weapon-related feats tended to get this requirement. Not because balance demanded this, but rather because their image of what each class was “supposed” to be like. But either option is pure speculation on my part, nothing more.
Regardless of what they were or were not thinking when they wrote the feat, the BAB +1 requirement on Weapon Finesse is not only unnecessary (just as it is not for Weapon Focus, Exotic Weapon Proficiency, or Quick Draw), but is also exceptionally problematic. That’s because, unlike the other feats I mention (with the possible exception of Quick Draw), Weapon Finesse is a build-defining feat – someone with Weapon Finesse is going to have low Strength and high Dexterity, and is going to be interested in melee combat. Thus, they need Weapon Finesse to function correctly.
Worse, this description, of a high-Dexterity, low-Strength melee combatant, does not describe most full-BAB classes – you can make a Dex-based fighter, certainly, and even a Dex-based barbarian with the right supplemental material, but these are not the norms for those classes. Fighters get proficiency with heavy armor and shields as a major class feature; you cannot use those on a Dex-based build. Paladins are worse – same proficiencies, but far fewer feats. And rangers are worst of all – their ability to ignore the Dexterity requirements of their Combat Style feats is a major class feature, one of the only ones that is both relatively unique and relatively useful for that class.
And then you have medium-BAB classes that, unlike the full-BAB classes, are often perfectly described as high-Dex, low-Str melee combatants. Monks and rogues wear light armor if they wear armor at all, have ¾ BAB, and find themselves in melee. Rogues in particular see a lot of advantage from high Dexterity: requirements for Two-Weapon Fighting feats, associated score for critical skills. These are characters who want to build around Weapon Finesse.
Which sucks if you are level 1, and have BAB +0. You are forced to sit through two whole levels of hideously poor accuracy, just so you can make the math work by taking Weapon Finesse at level 3. This doesn’t add anything to the game, it just makes those classes suffer needlessly.
The BAB +1 requirement on Weapon Finesse (and most other feats, honestly) is waived in the overwhelming majority of games I play in (the only games where it isn’t, that I can recall, are those in which no one wanted it and therefore didn’t bother asking the DM), and every single game I DM. In fact, most of the people I play with forget that it has that requirement. And our games are better for it. It allows us to play at low levels without being forced to suffer through unnecessary glitches like those caused by Weapon Finesse’s requirement.
I strongly recommend every DM to waive that requirement. I’d also recommend doing so on other, similar feats (certainly each of the ones I listed earlier). To not do so, I think, is a mistake. A mistake that the authors of Player’s Handbook made, true, so an understandable mistake, but a mistake nonetheless.
In fact, I would echo Thales Sarczuk’s suggestion, and strongly consider just making Weapon Finesse a free option.