The excellent science journalist Dan Vergano gives us his conversion story. Short version is he began as a member of the hippie-punching Cold Warrior tribe who reflexively disbelieved in climate change. His self-image as engineer opened the first crack though, when someone he trusted challenged him to check the data and he was educated enough to understand it. Not enough to be convinced though.
Convinced happened years later as a science journalist, when he applied the "nut test" to professed experts on both sides of the climate issue. The real scientist acknowledged the possibility of being wrong, while the denialist acknowledged none, and the issue was settled for Vergano.
I still find it somewhat perplexing to understand why people go unpersuaded for years by mountains of evidence and then finally change their minds, but this helps a little. Maybe we need a database about how to shake people from their mental frames.
Ironically, I think solutions work the opposite way for most people who spend little time on the issue and are persuaded by certainty and confidence. "The science is settled" is more persuasive to most people (and also true) than "our horde of scientists are still thinking rationally, while the tiny clique on the other side are controlled by irrational certainty and can't be trusted."
You have to understand your audience. Some years ago at my water district when it considered fluoridation, I was annoyed with the pro-fluoride proponents who claimed certainty that fluoridation was safe. The other side was even worse and I sided with fluoridation, but both were using the wrong arguments with me.
Side note: here's 10 minutes of pretty favorable coverage of the Pope's encyclical on Fox News Sunday. That type of coverage could help given the network it's on. Wallace trots out a version of the Lomborg nonsense about doing something to help the poor other than fixing climate, and I think the bishop flubs his response, but the back and forth is less important than the overall favorable coverage.