Starbases and Sensors

I don't think so. It's easier to stick a cargo ship loaded with sensors on top of it if you just want to see what's around the starbase. They become even more useless once you get Eye of the Galaxy since it only boost sensors on ships and not starbases or planets (shouldn't it be the other way around?...)

The sensors on the starbases will help remove the fog of war around crucial planets... not every time you get Eyes of the Universe... you may need some extra sensor range to keep an eye on your borders and mainly to help remove the fog of war in your territory.

they become even more useless once you get Eye of the Galaxy since it only boost sensors on ships and not starbases or planets (shouldn't it be the other way around?...)

Good point, I second that.

A patrol ship with maxed out sensors still seems like the more viable solution to me. How much range can you squeeze out of a starbase anyways?

sometimes a stationary object is more useful at guarding a border than a ship with maxed out sensors... the AI will be more inclined to target the Starbase than a wimpy scout... you can max out attack of a starbase and destroy a fleet of 10 maxed out ships with some of the best weapons with a starbase fitted one of those improvements in Starbase Fortifactaion III which improve attack by +20 I think...

Sensors probably aren't worth getting unless you:
1.) Don't get Eyes of the Universe
2.) Have starbases scattered around the perimeter of your civilization as a detterant and want to keep tabs on enemy ships
3.) Own a far-flung resource starbase and want to know what happens around it