Yes, nuclear power is at least a partial solution to climate change. But, to be fully effective, we need to convert all home heating and transportation to electricity-based, AND convert all electricity generation to nuclear-based. This will necessitate the construction of thousands of nuclear power plants around the world, a very expensive undertaking.
There are those who will argue that nuclear power is unsafe, and that the disposal of nuclear waste is a serious problem. But, if the consequences of climate change are half as bad predicted, then nuclear power is, by far, the lesser of two evils.
Warming of the climate system is unequivocal. Most of the observed increase in globally averaged temperatures since the mid-20th century is very likely (>90%) due to the observed increase in anthropogenic (human) greenhouse gas concentrations. (See IPCC Fourth assessment report).
Therefore, replacing carbon intensive processes such as fossil fuel electricity generation with low or zero carbon ones such as nuclear power could help mitigate climate change and its potential effects. However, supplies of uranium, might be insufficient to fuel a much larger number of reactors, although the IAEA claims we have enough high grade uranium ore at present rates of use for at least 85 years (see IAEA 2006).
Other nuclear technologies such as fast-breeder reactors which regenerate their own fuel, and thorium fuelled reactors, may have far greater potential. However, the economics and safety of these technologies still have to be fully demonstrated. In fact, economic considerations and political opposition due to the safety concerns of nuclear power has limited its widespread use, although some countries such as France continue to generate a large proportion of their electricity this way.