Coinbase is a hosted balance service, which means we manage your private keys for you. It is not possible to delete a crypto address from your Coinbase account. Deleting addresses from any wallet is highly discouraged since any funds sent to an address which has had its private key deleted will be lost forever.
If you're concerned about privacy, rest assured that Coinbase generates a new address for you each time you receive or send funds, so third-parties can not easily track your future transactions. Also, you can generate a new address for any of your Coinbase assets at any time here.