At Coinbase we go to great lengths to keep all of your sensitive information safe.
Account numbers and routing numbers are stored using bank level AES-256 encryption on our servers. In addition, all traffic goes over SSL to prevent third parties from eavesdropping on your connection. Employee access is heavily restricted and we require background checks on all employees.
For information on how Coinbase uses your linked bank account, please visit the Coinbase User Agreement, and the Coinbase Privacy Policy.