One way is to use another app to manage your dropbox that enables this functionality. I use Astro File Manager to generally manage the files on my device and this can also link directly to your Dropbox, Google Drive, SkyDrive, etc.
Astro gives you the additional option to "Open As", from which you can choose Text, Audio, Image or Video. Select "Text" and you get the usual list of apps that can open text files, including the DB Text Editor.
UPDATE: Unfortunately there appears to be a limit to how many files Astro FM is able to retrieve from a Dropbox folder. Somewhere between 1,000 and 1,100 files I get a "Connection Failed - Could not establish a connection to Dropbox" message when trying to browse the folder. 1,000 files (exactly) seems to be OK. There is no such limit with ordinary folders on the device.