How do I password protect a shared database that is being copied?

Hi Petas999.  If I understand correctly, you want to invite someone to use the base (to edit individual records) but you don't want them to be able to configure the tables or fields, or to share the base with others?  If so, you can accomplish this by assigning a permission level to your collaborator. 

Within your base, click "Share" and invite your collaborator to be an editor, which enables them to edit records and views but not to configure tables or fields.  Or you could invite them to be a commenter which would only allow them to comment on records, or read only where they could view but not edit or comment.  If you have a Pro or Enterprise plan, you can assign permissions to individual fields and tables, which gives you even more control of what is and isn't editable.  Finally, if you want your collaborator to have access to all the bases within your workspace, go to the Airtable homepage, find your workspace, and click Share from there.  

Here's more information about assigning permissions and what you can and can't do at each permission level: https://support.airtable.com/docs/airtable-permissions-overview  

If I share a database with someone and they duplicate it. Is it possible to disable editing or restrict further sharing before sharing?

No, there's currently no way to prevent a collaborator from duplicating a base.  

You could create a portal on top of Airtable that would enable users to view and edit certain data but restrict the ability to change fields, duplicate etc.