I know this is over a year old, but I wanted to leave this here if you or some other future Googler finds it useful.

Someone recently-ish gained access to my saved passwords in Google and was thus able to access my DigitalOcean account, reset my root password, and login via the console.

As a result I decided to implement this: https://github.com/shitchell/response-test

It detects if the login session is coming from a physical TTY (which the online console appears to be), and if so, presents a customizable challenge phrase/response (glorified second password). If they get it wrong, they’re dropped into a pseudo shell that mimics /bin/sh but with broken commands so that it appears not that they’ve been locked out, but that the system is just horribly broken.

Hello, all

The web console is intended to access your droplet in a case of emergency in order to sort out the issue. Most users manage their DigitalOcean Droplets using SSH, PuTTY, or a control panel they’ve installed themselves. However, changes to your network or service configuration files could leave you unable to log in remotely using these methods.

The Droplet console provides out-of-band access and is available regardless of your network settings. It emulates the access you would have if you were sitting down with a keyboard and monitor attached to the actual server. You can use this feature to log in and revert bad settings to regain normal access.

I will highly recommend to use a ssh-client in order to access and manage your droplet.

You can also check our docs on How to Connect to Droplets with SSH:

https://www.digitalocean.com/docs/droplets/how-to/connect-with-ssh/

There are two options at the bottom of the doc:
Connect with OpenSSH
and
Connect with PuTTY

Hope that this helps!
Regards,
Alex