Raid Eyes (and other AI Eyes) only trigger if there are 2x as many human ships as AI ships in the system. Ergo, you want to use things that pack a lot of whallop per ship. Starships are an obvious choice, but fleet ships with low ship caps can also work. There is a hotkey that lets you select ships of a particular mark; bringing just Mark IV bombers to a system might be a good way to deal with, say, fortresses in systems with an Eye.
If you bring enough ships into a system, all the guards in the system will leave their posts to come deal with your fleet. If you bring fewer ships, though, they'll stay at their posts and you can fight them in smaller batches. That does mean fighting next to a guard post, so make sure to keep any ships that post has bonuses against out of the way. Smaller raids can be very effective (and that's pretty much how you have to deal with AI eyes).
You can also use ships with tractor beams to drag ships into the next system over to be massacred by your full force. Riot Starships can pull these sorts of shenanigans and are always available.
Failing that, upgrade to the most recent beta version if you haven't, and then just drop a sabotage hacker in the system. It'll blow up the eye (possibly after some other stuff). If you haven't done much hacking, this can be a hilariously easy way to get rid of annoying things like Eyes and Fortresses.