Good afternoon & WELCOME to ECN.
A deteriorated jacket on SEU is an issue, and should be addressed. Wraping it with tape is IMHO not a 'fix'.
Your house inspector should pick this up on the home inspection & write up same.
Depending on what utility company area this is located in, the utility inspectors have the authority to deem it unsafe and inform the owner that replacement is required or the electrical service will be terminated by them.
As an AHJ, I would inform the HO that it is 'unsafe' and requires attention of a NJ Lic EC, along with a permit & inspection.
As it sounds (by your words) that it is old, you may also want to look at the grounding & bonding, and more than likely the panel.
This is not something that should be addressed by 'an electrical handyman', but a licensed professional.
