Technology has advanced a lot since Homer got out of the lead suit. A lot of nations just refuse to use said technology out of fears that there will be another Chernobyl (human error and poor maintenance) or Fukishima. Fukishima is a very real danger, but partially avoidable through contained systems [0] and a comparable threat for other power plant models)
Yes, you will eventually have some waste. But nowhere near as much as we used to. And there have been quite a few studies (and "common sense") to indicate that greenhouse gas emissions are MUCH MUCH lower [1] than most of the alternatives.
So it is a tradeoff. Obviously solar/wind/geothermal/hydro [2] are the best choices where feasible. But if you can't use those reliably in a manner that is able to provide required energy at all hours of the day (including peak usage), you are stuck with dirty approaches. And of those, nuclear is pretty gosh darned "clean".
[0] https://en.wikipedia.org/wiki/Small,_sealed,_transportable,_...
[1] http://www.world-nuclear.org/nuclear-basics/greenhouse-gas-e...
[2] Hydro actually tends to cause a lot of problems for wildlife and the like. Not aware of any drawbacks for the other three though