No, use of the API itself is actually entirely free. The API is built such that developers can create apps that (once in production mode) any Dropbox account can link to. The Dropbox accounts used with the app can be of any type (Basic, Plus, or Dropbox Business.)
It is only if the owners of any individual (or group of) accounts want to upgrade is there cost involved. You can see more information on pricing and plans here:
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?