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 US customers, you can verify your account in just a few minutes by entering your bank credentials. You may be familiar with this process if you've ever verified a bank account on Paypal or similar services. These banking credentials are never sent to Coinbase, but are shared with an integrated, trusted third-party, Plaid Technologies, Inc., to facilitate instant account verification. This service may store the provided credentials for verification purposes or use them on a per-transaction basis for fraud prevention, as well as help verify a sufficient balance is available to process your transaction.