If a nuclear reactor has a cooling failure, the fuel will eventually heat up and then melt down. To prevent this you have to keep water circulating around the reactor core.
This is the one simple problem.
There are now reactor designs that are passively fail-safe. So that even if all the pumps and everything stop, it will still not result in an explosion or otherwise serious release of radioactive material.
I don't know how much those sorts of designs were developed at the time Fukushima was commissioned though.
BWR reactors like Fukushima are all passive-safe in at least one way, since water moderates the neurons coming off the critical materials, slowing them enough to continue the reaction. If the water boils away, sure the thing will almost certainly melt but there won't be a runaway prompt-critical event like Chernobyl, and if the concrete "bathtub" around the core holds, there won't be much radioactive material released into the groundwater either.
From what I heard the concern at Fukushima was that the earthquake had compromised the integrity of the containment structure, so allowing the core to melt down was judged as too risky (I'm not sure if I agree with that decision, but of course my opinion in hindsight is not relevant). In fact the majority of radioactive material that was released in the incident might have leaked from the spent fuel storage, which was also damaged in the earthquake:
In fact Fukushima had a passive cooling system, but the valve to it was mostly closed when the power was lost and nobody knew how to open it or even check its state.
You could have a much safer design available, but if the 'less safe' design is more cost effective, and meets all specifications, it would still be the preferred design. This means that the any problem is with the specifications/requirements. The next question is whether the standard of safety which would require choosing the 'failsafe' (or at least failsafer) design would cause statistical murder.
This is the one simple problem.
There are now reactor designs that are passively fail-safe. So that even if all the pumps and everything stop, it will still not result in an explosion or otherwise serious release of radioactive material.
I don't know how much those sorts of designs were developed at the time Fukushima was commissioned though.