As far as I know, no.
The permissions you set on certain files are the same, no matter what user you (or the DigitalOcean Staff) is on.
So if you completely restrict access to a certain file, they can't open it just like you can't.
I'm pretty sure all they can do is check based on the information in your tickets, not particularly the files, to give you support.
You can find out more here.
Here's the line(s) that answer your question, though:
"DigitalOcean does not have access to its users’ server data. The backend is locked away from the users’ support staff and only engineering staff has access to the physical servers where users’ virtual machines reside. DigitalOcean does not store users’ passwords or private SSH keys. DigitalOcean also does not request user login information to their servers. DigitalOcean does not review or audit any user data."
I hope that answers your question.