Version 2.4.5.10 of the Dropbox app adds the ability to export files to SD card.
I have a Galaxy Note 3, and have installed this new version but when I select a file, and choose "Export" and "Save to Device", this is the menu I see:
Click image for larger version
Is there an option I need to enable or something else I need to do so that the "Internal Storage" and "SD Card" options appear?