The DO console is actually like having a mouse, keyboard, and monitor directly attached to the computer that is your droplet. As a result, it does not connect over the SSH protocol. The password used there will simply be the system root password for the droplet. If you do not have a password defined (meaning all passwords attempted fail, as is the default state when creating droplets with SSH keys), you can SSH in with PuTTY and type "passwd" to set a new root password.
If you created the droplet with an SSH key attached in our cloud panel, SSH login with root password will already be disabled, but you should always make sure the password is sufficiently strong.