Fiberoptic needs repeaters every couple hundred kilometers, so everything crossing any ocean is actively powered.
The damage to the cable would come from difference in potential at the ends of the cable. The cable might be at the ocean bottom but it could still get fried if you apply high voltage that could kill repeaters. And without those the cable is useless.
Basically, greatly simplifying, in a solar storm any long conductor might be subjected to high potential difference (potential difference == voltage). It does not matter if the middle is exposed but enough of it needs to be exposed at the ends. So a lot will depend on details of how the cable is actually terminated. And it should be possible to engineer it (ie terminations) so that it is safe from a storm.