Sink basin gasket leaking
Assuming by "fail" you just mean crack, deteriorate, or otherwise fall apart, just caulking will be fine. The gasket was probably a time-saver that came with the sink, but typical undermount installations just use caulk to seal the gap.

Trying to remove and replace the whole sink is a real project -- they're glued and clipped to the counter. I replaced my undermount sink, which looked much like yours, with a single-basin granite undermount... it took a full day and plenty of cursing.