Safety concerns are overhyped; nuclear power is safe
The waste from nuclear plants must be isolated for a fairly long time (but not as long as many think - only a few hundred years, if the used fuel is properly recycled). It can be handled with essentially no impact on the general public or the environment. Not so the millions of tons of waste each year from a coal-powered plant.
What about radiation from nuclear plants? Not a problem. It has been known for a long time that coal plants put more radioactivity into the atmosphere (from trace impurities in the coal) than nuclear plants do, even when more than 95 percent of the fly ash is precipitated. In the future, nuclear power will be the main workhorse. There's just no other way for humanity to get enough of the clean and safe power it will need over the next few thousand years. (Marsh and Stanford)

Nuclear power plants don’t cause leukemia.
Childhood leukemia rates are no higher near nuclear power plants than they are near organic farms. ‘Leukemia clusters’ are geographic areas where the rates of childhood leukemia appear to be higher than normal, but the definition is controversial because it ignores the fact that leukemia is actually several very different (and unrelated) diseases with different causes. Clusters tend to be found in isolated areas where there has been a recent influx of immigration – which hints at a virus. Men who work on nuclear submarines or in nuclear plants are no more likely to father children with leukemia (or any other disease) than workers in any other industry. (Johnston)

Nuclear energy does not result in illnesses; in fact, radiation is beneficial.
People who live in Ramsar, Iran are exposed to natural background radiation of 79,000 mrem per year, 5,266 times more than what the EPA’s 15-mrem/year radiation safety standard allows. The local river and its streams have a high concentration of radium, which is 15 times more radioactive than plutonium. Its 2,000 residents do not have an increased incidence of cancer, as the linear hypothesis would predict, and their life span is no different than that of other Iranians. Fortunately, for that resort, EPA regulations don’t apply there, or to people in Guarapari, Brazil, who get 17,500 mrem of radiation per year with no ill effect. Investigators have found that small doses of radiation have a stimulating and protective effect on cellular function. It stimulates immune system defenses, prevents oxidative DNA damage, and suppresses cancer. Despite what you might expect, atom bomb survivors in Nagasaki who received 1,000 to 19,000 mrem of radiation have a lower incidence of cancer, especially with regard to leukemia and colon cancer, than the non-irradiated control population. And it is turning out that Japan’s atom bomb survivors are living longer. (Miller)