The biggest problem with BAB is multiclassing. It makes it very difficult to play a fighter/mage if every level you sacrifice a +1 to attack. How do you attack with a sword if you can't hit the monster's defenses? If you were allowed to keep fighting lower level monsters it isn't really a problem, but you can't keep pace with a single classed characters.
The other problem was when melee characters (rogue, bard, cleric, etc.) were expected to attack the same defenses as a fighter, but simply couldn't keep pace. Backstab is nice, but if you can't hit the AC then you don't really have that ability. The cleric didn't really suffer for having a lower BAB because he could make up the +5 attack that he lost (and then some!) through spells, but the other classes couldn't.
A lot of the problem with BAB though would go away if you stop doing ever increasing monster defenses by level, and if you have more varied encounters. This is something that 2e did much better than 3e. In 2e you essentially had BAB (we called it THAC0) but the monster defenses only went to -10 (equivalent to +20) instead of +40. In the same adventure, you would find monsters with +4 to +15 to their AC. A rogue, or bard, or less than optimized character can still hit and be effective against all but the highest AC monsters, which is when the fighter got to shine.
In 3e and 4e however, all the monsters had the same amount of AC, which was designed to be challenging for the optimized fighter to hit. Which meant that the rogue always missed, that multiclassed characters missed, and the unbuffed cleric missed.
Hopefully with the flatter math, and the idea that you add more monsters to make weaker monsters challenging, we can go back to the encounter and adventure design that makes lower BAB melee characters viable again.