Apparently the handy and useful right click "copy graph" function is missing from Graph Builder in JMP 12. I tried using the "copy picture" function and it copies a blank (empty) graph.
If you use the same menu but rather than right clicking on your graph right click on the grey triangle (open close marker next to the title). Note: this is being passed on to you via helpful JMP tech support as I asked this question of them a week or so ago. Personally I am a selection tool person for copy pasting (using the keyboard s to get to the tool).
Craige referred to the 12.1 maintenance update which will be available this summer.
The 12.0.1 you've got now is an interim release that fixed only a single security issue.
Sorry for the confusion.