When I want to express he is sensible or insightful, can I say "he has a good eye." or "he has good eyes"?
Both of your phrases are correct but have different meanings.
Without any other context, your phrases literally mean
he has a good eye
he has one good eye
he has good eyes
both his eyes are good
Neither really express sensibility nor insightfulness which might be expressed as
he has a good understanding of
he has a good grasp of
Having an eye for something is to be able to appreciate the visual aesthetics of something
he has a good eye for design layout
he has a good eye for landscape painting
he has a good eye for catching continuity flaws in movies
this is usually shortened to
he has an eye for design layout