Nuclear is one of the safest sources of energy. Despite widespread public (mis)perception that Chernobyl and Fukushima killed thousands, and that radiation – and especially nuclear waste – poses mortal threats to all of humanity, the evidence is that it’s about as dangerous as wind, solar, and hydro, as shown by this graph from an examination of the issue by Oxford University's Our World In Data project.
Fukushima and TMI killed nobody, despite multiple reactors melting down. Radiation levels outside the Three Mile Island power plant did not exceed background levels. Even around Fukushima radiation levels were less than normal background levels in some populated parts of the world.
Chernobyl did kill people: 46 confirmed deaths, mostly from radiation sickness and thyroid cancer, maybe 4000 or so more, decades later, from cancers — if our more pessimistic assumptions about low level radiation are correct (which they may not be). And the evacuated zone, now largely free of human activity, is a teeming wildlife paradise.
But the Chernobyl accident involved a design of reactor which would never have been allowed to be built outside the former Soviet Union, built and operated in a culture which prevented learning from mistakes, and the accident occurred when most of its safety systems had been deliberately disabled in order to run an experiment on the reactor, which was being operated by a night shift crew who didn't properly understand what they were doing.
And that was the worst nuclear disaster ever.
Note that during the same period that we've had nuclear energy, aeroplane crashes have claimed tens of thousands of lives, but we don't have activists and politicians calling for air travel to be shut down, or aeroplanes to be replaced by airships. Nor are there calls for cars to be replaced by horses and carts, despite the carnage on our roads.
We are told that nuclear waste is an enormous problem and that it has to be isolated from the environment for hundreds of thousands of years. But the fact is that the dangerous isotopes are the shorter-lived ones, like Caesium 137, which have half-lives of a few decades and which decay to harmless background levels in a few centuries. Even without long term repositories the massively-thick stainless steel casks used to store spent fuel at power stations provide safe containment for decades, if not centuries, to come. And we can build safe, long-term repositories, as Finland is demonstrating and the UK and other countries are planning.
See also What about nuclear waste?
The IPCC notes that there is a legitimate concern that having nuclear power stations, and the technology to build, run and decommission them, might make it easier for a country to build nuclear weapons. In order to prevent this there are safeguards enforced by international treaty to prevent this happening. Countries which wish to take advantage of nuclear energy are obliged to submit to inspection of their nuclear facilities to ensure that they are not being misused for military purposes.
There are also technical barriers to proliferation from civil nuclear energy to nuclear weapons. Uranium fuel for nuclear power stations is far too poorly enriched to use for weapons, and modern power station reactors are quite unsuitable for breeding plutonium for weapons. Many countries with nuclear energy simply buy fuel from suppliers in other countries. In countries which enrich their own uranium enrichment facilities are subject to particularly stringent scrutiny to ensure that they are only enriching to low levels suitable for reactors. Since reactor grade fuel is no more than 5% concentrated and weapons require a concentration of around 90% it is hard to disguise enrichment for weapons purposes. History shows that those nuclear-armed countries which have civil nuclear power acquired material for their weapons either before developing civil power or separately from it (some, like India and Pakistan, through research reactors). Nuclear armed Israel and North Korea have not developed civil nuclear power at all.