Brand should have its base damage nerfed and its scalings buffed
I think there are more problematic champions than Brand currently, and he isn't the first champion that I would think if someone asked "what champion would you nerf in this patch?" (I am speaking of the amount of Brand threads that have been posted recently)
With that said, I think Brand's main problem is that his base damage is too high, yet he doesn't scale nearly as well with AP. This means he can be very useful as a support, because supports tend to have lower gold income, which will not be a problem for a champion that has high base damages. It also means that he isn't that amazing as the team's main AP carry, because he doesn't scale as well as he should.
It also means that, even when behind and/or having few AP items, he can destroy you.
What should Riot do to fix Brand's problem? The solution is obvious; shift some power from his bases to his scalings. His passive was buffed to 4% of the enemy's maximum health to compensate for DFT being gone, when what Riot should have done is to increase the AP scaling from the explosion when you manage to hit three stacks with Brand's abilities. This would reward him more for comboing the enemy, not just for throwing a spell.
Also. 35% AP scaling on E? 25% AP scaling on the ult? No wonder why Brand is rarely played mid, he isn't rewarded enough for building AP and having a higher gold income.
Reducing Brand's base damage would nerf him as a support, since he would be less useful when behind, which isn't what you want on a support (a support should be useful even when behind, in contrast to a carry, which should scale well, but you can shut down and prevent him from scaling to begin with).
TL;DR: Brand feels OP currently because he has insane base damage, which allows him to heavily damage you even when you are behind. This also makes him very strong as a support. Brand was intended to go mid, but he isn't rewarded enough for building AP to allow him to be more than a niche pick in mid.