Dropbox's website suggests that they store files using some sort of AES-256 encryption and they cannot read them: https://www.dropbox.com/help/27 I wouldn't make the claim that they are more (or as) secure as Tarsnap (especially since there is no page as detailed as http://www.tarsnap.com/crypto.html on Dropbox's website), but this suggests that files on Dropbox are not readable by the company.
If you forget your password, can Dropbox reset it for you? If they reset your password for you, can you still read your data? If the answer to both of these is "yes", then dropbox can read your data without you providing your password.
(Honest questions -- I don't use Dropbox, so don't know.)
