I would recommend using your Blockchain.info wallet which supports both BTC and BCH but gives you full control over your private keys as opposed to Coinbase which controls your private keys. To import your coins into your Blockchain.info wallet from your paper wallet do the following:
- Log into your wallet
- Click “Settings” on the left hand side of the screen
- Click “Addresses” from the dropdown list
- On this screen you can select either “Bitcoin” or “Bitcoin Cash”. Select “Bitcoin”
- Click on “Import Bitcoin Address”
- Enter in your private key from your paper wallet and this will “sweep” all your Bitcoin into your Blockchain.info wallet.
The same private key also controls your Bitcoin Cash since you had them in the paper wallet pre-fork. I’m not sure if the above automatically imports your Bitcoin Cash into the Bitcoin Cash section of your wallet or if you have to repeat the process under the Bitcoin Cash option. Either way your coins are still under your control. Now you can start transacting with them on their respective blockchains. If you aren’t planning on doing any transactions with then, I would just keep them in the paper wallet as this does not expose your private keys to the internet for potential hackers. Just make a backup copy or two of your paper wallet and keep in a totally separate (off-site) secure location in case of catastrophic loss to the original paper.