Reposting a comment I read on the internet once and thought was the best answer coming from a pro-nuclear person:

> I've said this before and I'll say it as many times as is needed.

> The problem with the waste (and nuclear and radioactivity in general) is the messaging coming from those who do understand it. That is what solidifies misconceptions and fear, just as much as what anti-nuclear activists fearmongering may contribute.

> The average person believes that the mere existence of nuclear waste is an existential threat. That it's like a monster trying to break free from its container, and if it does, or if someone lets it out, it will wreak absolute devastation on the biosphere of potentially the entire Earth.

> You cannot appease or dispel such fears by insisting about how seriously waste is being handled and how we've gotten much better at it and that we've reduced the chances of accidents and contamination by this and that.

> Those arguments do not matter when the belief is that the threat is world-ending. People do not believe that the chances of accidents are zero - nor should they be expected to.

> What you need to tell people is that their inherent belief is wrong. Like it's absolutely, completely wrong on so many levels that it isn't even in the same ballpark as ludicrous. I know, I know, they will probably not believe you, at least it will take a lot of time for them to start believing you. But you have to say it nevertheless. Stop reinforcing their fears by always talking about how safe you've made the process, that will never work. Never.

This talks about nuclear waste but I think it applies even more to nuclear accidents. At least it applied to how I saw nuclear power before I actually decided to learn about it (after watching the Chernobyl series lol !)