My cousin has a kid that I would really, really rather not babysit.
Why not. What makes you against babysitting the kid?
I'd rather not be responsible for other people's  children.
It is not fair for your cousin to expect you to want to babysit their child. However, if you agreed to babysitting the kid then you should follow through with what you said you were going to do.
It's not my job. Just because they're my cousin doesn't mean I have to babysit for them.
It definitely isn't your job, and you shouldn't be expected to do it just because of the family relationship. However, if you made a commitment, you need to follow through with it, otherwise you make yourself unreliable and untrustworthy.