One thing to note: For non-sandboxed apps like Dropbox, the Accessibility API permissions don't really decrease security by a lot (in my opinion).
Most bad things can be done without the Accessibility API, e.g. apps can act as key loggers, take screenshots, encrypt all files your user can access, upload arbitrary things (unless you have a firewall enabled), synthesize mouse & keyboard events etc.
The Accessibility API makes some of those things easier, but if someone really wanted to attack you, he wouldn't need the Accessibility API.
For sandboxed apps the situation is quite different, because the Accessibility API would allow those apps to break out of the sandbox.
But of course Dropbox should have asked the user...
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