Another layer of security for your Dropbox account (two-step verification) (blog.dropbox.com)

Dual authentication only occurs when you would normally be asked for a password, so assuming you have already linked your computer, you won't need to authenticate again.

No, you'll only get prompted on desktop/mobile the first time you link your account. If you already have a linked desktop or mobile app, it will continue to work without a prompt.

You can also see all of your currently linked devices here: https://www.dropbox.com/account/security