I don't want dropbox remembering my logon password. It should have to be typed every time I log into it. I don't see where to "do not remember me." Is this possible?
it saves your credential information. To "forget" your account on your machine -- right click the Dropbox icon in the task tray --> Preferences --? Account --> Unlink This Computer.
Then exit dropbox (right click --> Exit) then relaunch the app and it should be like when it was first installed (Do you have an account? Y/N - wizard setup).
Dropbox assumes you sync it with your own computer. In this case you can simply password-protect your computer/system account and change Dropbox folder permissions.
If you're using Dropbox on a public computer then I don't know why you'd want to sync entire Dropbox. You can simply download the files you need through their website and it won't remember you.