It has never been a problem in the first place. At least not a technical problem. Sure, it is nasty stuff, to be treated with respect, but not worse than toxic chemicals we regularly deal with. It is also potentially useful stuff: rare, exotic matter that gives off energy. Compare with coal plants that also produce nuclear waste (there are radioisotopes in coal), but that waste is dumped in the air you breathe.
> Are old powerplants properly decommissioned and replaced with new ones when the time comes?
The are properly decommissioned, and it is really expensive, and that's indeed a real problem with nuclear power: it is expensive. But the thing is: nuclear plants don't have an expiration date like bottles of milk, you could potentially run them forever with regular maintenance. The reason we don't do that is that is that over time, maintenance becomes more and more expensive, parts become obsolete and stop being produced, etc... At some point it is cheaper to build a new, better plant and decommission the old one instead of spending a fortune on obsolete parts and retrofitting.
> Can we really claim to be able to handle worst case scenarios well (e.g. Fukushima)?
No, they wouldn't be worst case scenarios if we could, but we are doing our best. In the end, even with Chernobyl, nuclear power is still one of the safest per unit of energy produced. It is no excuse, Fukushima shouldn't have happened, but if the only response to an accident was to stop everything, we wouldn't do much.