Looks like it doesn't retain the password, it just uses it to grant itself access to a config database which it can then modify at will later. Not that that is much better since accessibility permissions are kind of the keys to the kingdom and being able to modify the contents of that database is a highly restricted operation for a reason.

If you read the entire article he's says what happens if you don't give it your admin password

Yeah I didn't think that there was a definitive answer in the article, kind of says that dropbox will just keep asking for it on boot. I was asking to see if anyone knew what APIs dropbox might be hooking into, but @szhu in a post above has kind of confirmed that nothing really happens. I took a look at this 2009 post. Interesting.

pretty disappointing really.