This Seeking Tech guide explains how you can save Cid in Final Fantasy VI.
In order to save his life, you need to keep feeding him fish found on the shores to the south.
However, if you feed him the wrong type of fish, then it may end up worsening his health.
Instead, you should only catch fast moving fish. If you don’t see any fish that fit this description, then you need to talk to Cid again to reset the fish that appears on the shore.
Keep feeding him the fast-moving fish until he gets healthier. His dialogue should become more optimistic over time if you are on the right track. Eventually, Cid will fully recover.