Sharing a Dropbox link directly to the clipboard, without having a browser pop open?
No, you can't just get the link in clipboard, which is much safer for you. With Dropbox's version 2.0.5, you can easily share them through the toolbar.

There's a second way of doing it, being a lot better, which is:

Finder > Dropbox > public (move your public file here) > right click > dropbox > Copy public Link