I don't think it has anything to do with the bombings in WWII. For one nuclear power does not equal nuclear weapons. They are very different things, a nuclear power reactor could never detonate like a nuclear weapon. I think it is mostly due to the accident and probably even more so with the failings of TEPCO and the Japanese government to handle the situation appropriately. Also people are scared of things they don't understand, and the media hype doesn't help. The more you understand about the way nuclear power works the less you fear it, though that doesn't mean you don't respect the dangers involved. I don't fear nuclear power but I also think that we need to keep researching and developing other options. My opposition to it has more to do with dealing with the waste that is produced. I don't approve of just burying it under some mountain or keeping it in dry casks or in cooling pools. There are things we could do with it, for example there is a type of reactor that we could use that spent fuel in and it would "burn" up most of it while extracting a considerable amount of power at the same time. The waste from that type of reactor is substantially smaller in amount and also far less radio-active. I think I remember reading that the waste would have a half-life of ~300 years, far better then the ~10,000 years of the waste from current reactors. This type of reactor would not only provide power but also allow us to use the current stockpile of radioactive waste as fuel. It of course has other issues to be considered, I think one was that it was liquid sodium cooled. The reason we haven't seen these reactors being built is the same reason other modern reactor designs haven't been built, namely public opinion (not in my back yard) and cost.
I don't have time now but I will look up that reactor type and post a link to more info about it. I want to say it was an IFR type but I could be mistaken about that. I will try to post an update later tonight with a Wikipedia link or something.