how do I connect a new wireless keyboard to a 2013 MAC that is password protected?
I wrote the whole question in the subject line
iMac 27″, macOS 10.13
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If the keyboard is an Apple Magic Keyboard rechargeable, then connect the keyboard to the Mac with the USB-Lightning charging cable that came with it. (Maybe not, if it's a USB-C cable.) Or use the USB-Lightning sync cable that an iPad or iPhone use. The kbd will behave as a wired keyboard.
OR... use a wired USB keyboard to login to the Mac and then proceed to pair the wireless kbd with the Mac after login.