Hi
Generally it’s the outer sleeve of the cable that fails.
If the internal wires are ok (jackets are not damaged), the best way to fix it is to use a drop of hot glue and then shrink tube. It’s a far more permanent (and elegant) solution than electrical tape. The hot glue helps to make sure that the internal wires remain put and the shrink tube provides better insulation that electrical tape because once in place, it remains there forever. The electric tape in this context has a tendency to unravel.
If the sleeves of the internal wires are damaged, then stop using it. It can cause a short-circuit and damage either the power adapter, the logic board or both. Sometimes fuses can be replaced, but it’s a costly process which far exceeds the cost of a new adapter.
Cheers!
Bogdan