How to password protect printing on a HP Officejet?

Set up a VPN and make the printer visible only inside it.

This way, every device connected to the VPN will be able to see and use the printer. Outside the VPN, the printer or any other network resources won't be accessible.

I'm always recommending to never expose these network devices publicly. Exposure might look handy at the beginning but this does not go together with privacy expectation you expressed in your comment above.

I know this is not a direct answer, but it actually "password-protects" your network including your printer.

Your HP ePrint account can be set to accept print documents only from some specified email accounts. Currently it may be set to accept from everyone.

This is a weak security measure, but may be enough to prevent 'Serpent People in New York' from printing again.

References :

• HP Connected
• HP Printers - Turning On Web Services
• Introduction to HP Web Services (video)