Alpha Centauri B is about 90% of the mass of the Sun and roughly half its luminosity. so it wouldn't be quite as bad as that. In addition, if the planet was in the Goldilocks zone for A, it would also be at least ten times closer to A than B. (The minimum distance between A and B is about 11AU.)
So the gravitational influence of B on an Earth-like planet of A would be a maximum of 1/100 of the main sun's gravity - comparable to solar tides on Earth - and the maximum extra light would be 1/200. Which would make for bright nights, no doubt, when B was at its closest, but I don't really think this would be a problem - especially since, of course, life on a planet of A would have evolved under these conditions.
In the case of a planet of B, IMHO the gravitational situation would be actually less disturbed - because a planet of B would have to be closer to its sun because of the lower luminosity. However, the light of A would then be something like 1/50 that of B at most, which is getting significant - but, again, life would have evolved under those conditions.