There seem several possibilities.
Are you using Adobe Reader instead of Adobe Acrobat?
Are you making a PDF/A file? (It will say so under the menu bar)
Ah, yes, I'm using Adobe Reader! That makes sense. Thanks so much for your reply. At the moment, I scan Word documents into my computer and they're automatically converted to PDFs. If I had Adobe Acrobat on my computer, and scanned in documents, would they open in Reader automatically or could I open them in Acrobat and therefore add password protection?