I have simply followed the online instructions for Linux, on the remote server. That is, I've run the following commands:
cd ~ && wget -O - "https://www.dropbox.com/download?plat=lnx.x86_64" | tar xzf -
I was re-directed to the Dropbox website to accept linking my account with the "new" (remote) computer, which I did. A Dropbox folder appeared on the remote computer, and I can see all my Dropbox folders but only a handful of files.
I've been able to interact with Dropbox using the Nautilus package. I've excluded some folders with the hope that they won't sync (I only need to access a few files), and I can check the status of Dropbox, which appears "Syncing...". I was hoping to speed up the syncing process so that I can read the files I need from Dropbox directly on the server, and save the output of my work directly to Dropbox. That is, I want to use Dropbox to exchange files from the remote server to my local computer.