Step 1: Add the Dropbox repository to the Linux Mint package source file. The most suitable place in my opinion will be the application specific setup list ie /etc/apt/source.list.d/dropbox.list
# cd /etc/apt/source.list.d/
# touch dropbox.list
# vim dropbox.list
Go to http://www.getdropbox.com/download and cut and paste the repository information for Ubuntu 8.10 into /etc/apt/source.list.d/dropbox.list
deb http://linux.getdropbox.com/ubuntu intrepid mainStep 2: Update the Repository
deb-src http://linux.getdropbox.com/ubuntu intrepid main
# apt-get update
Step 3: Install Dropbox
# apt-get install nautilus-dropbox
Step 4: Restart Nautilus to start Dropbox
# killall nautilusStep 5: Menu > Applications > Dropbox
The rest is as straight-forward as you can get.