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There is currently no way in the Websites+Marketing plan to make private - or password protect - only a portion of a page or function. However, by creating the private page, then assigning permissions to members of your choice, you can control who accesses the page, and thus, the download. The other option is as you suggested, to leave the page open, but password protect your downloaded files, using the program that created them.
Supporting you at godaddy.com/help/contact-us