Last I checked, server-side Rust usage at Dropbox is reserved for the very bottom of the stack, for the bits that are performance-sensitive enough that the alternative would have demanded they be written in C++. Apparently there's a significant amount of Rust in Dropbox's Windows client as well, though I don't know the story there...
We're making a pretty big bet on Rust in the client (if you'd call it a bet, I don't see a scenario where we walk this back). The majority of new Sync code in the client (read: not UI) is being written in Rust and it's been a blast so far, not to mention much easier to reason about correctness which is really what we care about.
