How do I stop people from sharing a folder I've shared with them?

Thanks Walter for the helpful reply to @jezhawk.  I just want to add one thing.

If I understand your use case correctly, you want people to be able to access a shared folder of content, but not be able to share it further.  In other words, you want to limit sharing of your content to a specific set of people.  

Dropbox does allow you to do this by sharing directly with specific people:
1.  Open sharing settings from the share model (Share this folder window where you enter emails or copy link).
2. Under Folder Settings, set Manage Access so only the owner (you) can add people to the folder.

3. Under Link for Viewing Settings, either delete the link or set Who has Access to Only People Invited.  
4. Now you can invite people by email to the folder.

This will limit access to the folder only to those emails you have invited.  They will need to sign into an associated dropbox account to confirm they have been invited.  They will not be able to invite others to the folder, and if they attempt to create or copy a link, that link will also limit access to only people invited.

Hope that helps!