Here are his own words in an interview in 2008;
I said most people have the idea that Costner is a Republican, but he says he’s an independent.
“I grew up in a Republican household, and I heard [my] father go, ‘Damn that Kennedy,’ and when you’re 12 years old, you’re probably thinking, if you love your father, yeah, damn that Kennedy. My family was very staunchly Republican. I had to evolve and that took time for myself. I moved very quickly as a 20-year-old. I have very conservative roots, but my evolution allowed me to make ‘JFK’ and ‘13 Days’ and those kinds of movies. I’ve always felt too limited by both parties.”
I read a bit more about him and have to say that after reading several statements from him, I cannot say his is conservative. Indeed, he probably is not. Sad really, thought he was a better man.