The simplest way to change public opinion is manually. Skynet seems like an adequate solution to me.

The biggest problem with the movies, besides the inconsistencies as to whether causality is changeable or not, is why Skynet bothers dealing with the humans once it's broken their ability to prevent it from launching itself into space. Sending a single self-replicating seed factory to the Moon is what a reasonable AI would do.

The Terminator movies exploit the primal human fear of being exterminated by a rival tribe, putting AI in the role once filled by extraterrestrials: outsiders with great power who want to destroy all of 'us'. The pattern is tedious and predictable.