I would think that the problems that Japan are still trying to clear up should make any nation think twice about having nuclear power. Given the way that we appear to be suffering more natural disasters such as earthquakes and tsunamis, the way that such nuclear power, however "safe" its proponents argue it to be it brought into question. In addition, considering the way that Australia is an island--albeit a very big one--would potentially make it more vulnerable to any tsunamis.
The most important reasons why (one can argue) Australia should not use nuclear power are environmental. They have to do with A) the problem of nuclear waste storage and B) with the danger of disasters such as the one that struck the Japanese nuclear power plant earlier this spring.
Spent nuclear fuel remains radioactive for centuries. There is no good way to store it that is completely safe. It will have the potential to cause environmental damage or even to be stolen by terrorists. In addition, nuclear power plants can cause widespread damage in the event of a disaster. Australians might think that the disaster at Fukushima cannot happen in Australia, but there are other ways that things can go wrong at a nuclear plant. It may not be likely that anything will go wrong, but the consequences of a disaster are huge.