I'm not going to support my pregnant cousin
Why? What do you mean by support?
Giving her rides to the doctor; she makes me be financially responsible for her.
It's okay to set boundaries around the support you give your cousin, but it's good to help her out.
I'm not responsible for her anymore; it's her fault she's pregnant, I don't want to be involved.
You should talk to your cousin about what you can and can't do. It's not okay to  just abandon family, but you're allowed to have your own boundaries.