The Dropbox Business API offers a variety of methods for managing members of a Dropbox Business team, but it doesn't offer the ability to programmatically reset a member's password in particular. I'll send it along as a feature request.
(By the way, for reference, the Dropbox API isn't considered a "REST" API. It offers an RPC interface using JSON over HTTPS and doesn't follow REST conventions.)
We love to learn from the educators who use Dropbox. Whether you teach kids, teens, adults or a combination of all three, we want to know what apps and integrations you use with Dropbox to help with teaching. Which of the ones below is your favorite, or most used tool?