Would a solar flare induce enough current/voltage on these cable to actually cause a problem? Also by their nature of being under an ocean of conductive salt water wouldn’t there be some form of grounding providing protection?
So the effects of salt water are pretty strong, but not impossible to overcome. Extremely Long Wave radio systems are actually in use to provide submerged submarines with signals indicating they need to surface and use higher bandwidth communication channels. https://en.wikipedia.org/wiki/Communication_with_submarines
Now since there’s a level of RF penetration, you’ll get some antenna effects inducing fluctuating currents which will vary along the cable depending on the depth.
On top of that, the largest CME events can and do affect the earths magnetic field, squashing and warping it. This “moves” the earths magnetic field, and electromagnetism 101 is “conductor + magnetic field + one of these two things moving = induced currents”. So the movement of the earths magnetic field will be able to induce current all the way along the cable regardless of depth, unaffected by the seawater.
How significant the induced current and voltage are... and how vulnerable the powered repeaters and head end equipment are... is something I’ll defer to appropriate experts on.