Hash is 6 characters long, characters are alphanumeric (a-zA-Z0-9). So that makes:
(26 * 2 + 10) ** 6 => 5.6E10
That looks like a big number, but it isn't. Because at the scale of dropbox there will be 10 million links out there in no time. So then the math goes: ( (26 * 2 + 10) ** 6 ) / 10_000_000 => 5680
So you have to make only a few thousand guesses to get a random file from another user. I'd say that's not very secure.Note that the links redirect to a page with a far longer (and presumably far more secure) hash code. Any time when you see short hash -> longer hash alarm bells should go off.
I'm assuming the share links last forever. If the share links would last only 24 hours then system looks pretty safe.
Anyway, this is only my first impression. I might very well be wrong. Either way I think it's pretty silly to give up so much entropy to get a prettier URL. Why not just use the complete 128bit hash?
Anything you give a url to you should consider to be public anyway.
Can you tell us how to change that?
Remove linked items here: https://www.dropbox.com/share#tab:linked-items
So, if you feel like "I wanna share this folder to the wild world, now!", just do it! You can change your mind at anytime later and the folder will be no longer accessible. Very intensive, careful design & implementation!