Existing nuclear reactors are quite safe (aside from nuclear proliferation risk), and new designs using things like accident-tolerant fuels and molten-salt thorium reactors could improve this further, but safety isn't the main problem with nuclear power, and in fact even public opinion isn't; the main problem is the price of the generated energy. And there are no designs in the works that attempt to solve this problem.
With the exception of betavoltaic batteries, nuclear reactors provide energy by driving some kind of heat engine, and the cheapest large heat engines are steam engines. (Specifically, supercritical steam turbines, using Charles Parsons's 1884 turbine design, but with better metals, better bearings, and hotter steam.) These are the same engines that convert heat into power in coal power station; a coal power station is basically a nuclear power station without the nuclear reactor and with a lot more global warming. So, no matter what design improvements come along for nuclear power, we can't expect nuclear power stations to be cheaper to build than coal power plants.
https://www.solarserver.de/service-tools/photovoltaik-preisi... says that low-cost photovoltaic panels are now €0.19 per peak watt, unchanged since January. Last year they fell by 30%, but apparently the PV panel industry has formed some kind of cartel to keep prices high, like the DRAM price-fixing racket in the early 1990s — they announced last year that module prices would stop falling, and so far, that's holding true: https://news.ycombinator.com/item?id=19002696. (Chuck's skepticism in that comment that the cartel would actually be able to raise prices seems to have been borne out thus far.) So we're stuck with €0.19 per peak watt until the cartel breaks down and pricing reverts to the long-term learning curve in a year or three.
But, in much of the world, that makes installing photovoltaic capacity already far cheaper than building new coal power plants, not counting the cost of the coal. And that means it's also cheaper than the cost of building new nuclear power plants.
Whether solar power is cheaper or not in a particular place depends on the "capacity factor", the ratio between a power station's average output and its peak output or capacity. For example, solar panels don't produce power at night, and they produce a lot less when it's cloudy. In the US, this averages about 25% for utility-scale PV, with 27% in Arizona and only 15.5% in Maine. That means that an average watt of solar power costs €0.70 (US$0.79) in Arizona but €1.23 (US$1.37) in Maine. More equatorial and less cloudy places like the Sahara and the Atacama have even higher capacity factors, so solar energy is even cheaper there: https://www.forbes.com/sites/quora/2016/09/22/we-could-power... (Note that the pricing in that article is from 2016, when panels cost €0.53 per peak watt, two and a half times what they cost today.)
(Those prices are per watt, not per watt-hour. How much it adds up to per watt-hour depends on the discount rate you use and what lifetime you use. Crystalline silicon solar panels, which are the kind we use, last basically forever, although they lose about 20% of their initial power output after a decade or three. If you just divide the above numbers by 15 years, US$0.79/W comes out to US$1.67 per GJ, or in folk units, 0.6¢ per kilowatt hour. That doesn't include an inverter, batteries, or the rooftop panel installer's worker's-comp insurance, much less long-distance power transmission.)
And that's why, in April, Solarpack signed a contract with Chile to build a 120-megawatt PV plant in the Atacama, selling the energy at US$29.10 per megawatt-hour: https://www.zmescience.com/ecology/climate/cheapest-solar-po... This will lower the cost of electrical energy to businesses by 25% starting in 2021, and this price does not include any subsidies. 2.9¢/kWh (US$8.08 per gigajoule, to use SI units) is cheap enough that no coal or nuclear power station has ever reached that average price — and, unless we find a cheaper way to build steam engines or other heat engines, none ever will.
The intermittency of solar power — much bruited by nuclear-energy promoters — will eventually start to become a problem as more and more coal and nuclear baseload plants are decommissioned, but at least for several years, batteries are cheap enough to solve that problem, and they're getting cheaper, too. Gas peaker plants are inefficient, but they're also quick to build pretty cheap, and they still emit less greenhouse gases and carcinogens than coal. Through the 2020s, we'll see more combined-cycle gas peakers, more utility-scale storage projects, and more long-distance transmission projects, including HVDC. We'll also see more demand response, and in the end I think that will play a bigger role in solving the problem than utility-scale storage — ice-reservoir chiller systems, phase-change reservoir space heating and process heating, daytime EV charging, and simply shutting down energy-intensive production lines at night. If solar panels get sufficiently cheap, even long-distance transmission infrastructure and rural distribution will cease to be profitable except in extreme cases.