You can't, really. You can close your account, and you can email their support and ask them to delete all the information they hold on you, but you have no guarantee that they won't just say they've deleted it but actually continue to store it. Given how generally shady Coinbase's behaviour is, I think it is more than likely that they will continue to hold your information.
IIRC, Coinbase (or any other licensed/registered exchange) legally isn't allowed to delete all user data they store on their servers.
The thing with verifying yourself at whatever exchange is that you should consider your private information to be compromised for ever. There is literally nothing you can do about it. Even the exchange's word that they did delete all your data is worthless. In case of Coinbase, they have violated their own terms to list XRP, and they have participated in insider trading more than once, so that's shadiness on top of more shadiness.
Down with Coinbase.