There are some people who are always going to tune out a message, especially if the message is controversial. As a generality, most people don’t like change. They don’t even want to hear about change! So for many people a subject like climate change is threatening, and they react defensively by relegating it to “just politics.” Then they can rationalize not listening.
I don’t know the best way to inform in these circumstances, and there probably is no “one size fits all” approach. Messages that speak to what a particular audience is interested in may succeed, especially if the message is given impartially and even somewhat gently/compassionately (rather than hitting people over the head with the information). Maybe figuring out what’s important to a particular community or group, and how climate change will likely affect that particular thing of importance. I also think it helps to combine a negative message (the many downsides of global warming are a naturally negative message) with what positive steps can be taken to learn more, do more, etc — but not in a way that’s overwhelming (which makes the listener feel they can’t possibly make a difference so why try).
That said, I also think there are times to speak strongly regardless of how people react. Some people will take note, listen, and act. Both approaches can work depending on the timing and the audience.