There is really no objective way to answer this question. The answer to the question depends on how one defines “responsible.”
Nuclear power is clearly an alternative to the fossil fuels that we use today. Each type of fuel has its pros and cons. Each can be seen as responsible or irresponsible.
Fossil fuels are very unlikely to cause tremendous disasters to human beings in the short term. They will not melt down and spew radiation across the countryside the way that Chernobyl or the Fukushima plant did. However, they will almost certainly cause broader damage to the environment as a whole over the long term. The effects of global warming will have a greater impact on much more of the world than a nuclear accident is likely to have.
Nuclear energy, by contrast, is not likely to cause long term damage to the planet as a whole. Instead, it carries with it the threat of catastrophic accidents on a relatively small scale. Nuclear energy could help to prevent global warming, but it could also end up creating more accidents of the sort Japan suffered in 2011.
So which is more responsible? Is it more responsible to use fossil fuels and harm the entire planet in the long term or is it more responsible to use nuclear power and create the possibility of terrible, if somewhat localized disasters in the short term. This is one issue with which society must deal as we try to figure out how to provide ourselves with the energy we need.