I did this. My Computer restarted during the night. I think my droplets are still mining and likely do not need any supervision until the time comes to shut them down?
EDIT: It seems I must keep these windows open for the servers to keep mining. Oh well.
Also, is there a way to change the root password? This must be a totally noob question for all you Linux pros out there, but I am totally new to this...
And thanks!!!

Use the "screen" command.
http://www.gnu.org/software/screen/manual/screen.htmlHere's the gist of it, I'm not going into detail as that's covered in a plethora of howto's splattered all over the intertubes....
If you ssh into a server and run minerd, then your ssh session disconnects due to network problems, the server times out your login and terminates all running processes associated with it. As well it should, that's just sensible housekeeping.
Instead, ssh into your server, call up "screen", then run your minerd in the screen session. If your ssh session is interrupted, no matter, the miner process carries on within screen and you can reconnect to it when you next establish an ssh session.
screen is available in pretty much all common Linux distros.
screen is just one of several ways to kick tasks into the background and disown them from your current login session such that they don't get killed if you get disconnected. This problem was solved a long time before mining came along.
Hit up your favourite search engine with something like "Linux how to use screen" - you'll have to learn a few commands to disconnect, list or reconnect screens, but they kinda become auto-pilot once you have more than one remote machine.
GL