Printing with Safari Version 12.1

Best easy solutions for now are:

1. Use the 'Export PDF' button in Safari. Open the PDF in Preview and print from there.
2. Use for printjobs another browser like Firefox or Chrome.

A complete clean install of Mac OS is also an option but a lot of work and no guarantee that the printbutton in Safari 12.1 starts working again.

Have you tried disabling all the extensions in Safari/Preferences/Extensions?

Try setting up another admin user account to see if the same problem continues.

Isolating an issue by using another user account 

I think that I'm seeing the same problem. Immediately apparent after upgrading to Safari 12.1. This is about the action of the File>Print menu option. This is not about being able to physically print from the computer. I see that the File>Print action works (it brings up the usual printing pop-up) for a few minutes after restarting Safari, then it stops. This sure looks like a software regression.

I am also running the latest High Sierra. Not Mojave.

This started happening right after updating to Safari 12.1. In less than a day after that update there was a security update, 2019-002. That could have caused this, but it seems likely that it is problem in Safari.

       Startup in Safe Mode.

       Restart normally by clicking the “Restart” button at the bottom.

       Start up in Safe Mode. https://support.apple.com/en-us/HT201262

Try going to System Preferences/Printers & Scanners, then control - click in the left panel, and reset the printing system. Add back the printer.

Does command - P work?

According to AppleHint, To make print function works correctly with Safari, just simple go to System Preferences > Printers & Scanners and then reset the current added printer system.