Ned was smart enough to know that tellling one person is too great a risk cause people tell people secrets. The only way he could ensure it stayed one is by telling nobody. I really respect him for that.

Catelynn treating Jon like shit also allowed the lie to live on. In everybody's eyes he is a bastard and it's not common for a bastard to be raised in the house with the other kids...however, the fact that Ned's wife has an issue with makes sense. It just adds that extra layer to the lie thereby adding that extra layer of protection for Jon.