Well I just tried to find the thread with pictures on ZX-12R.org but evidently they went to a new fourm and the posts are there but the pictures are not.
You can read this thread, and then you can put "stator" into the search on the site and read all the countless posts about it.
Stator Connector Melted - Kawasaki World
I don't know where you live but if you were close to Pittsburgh, PA I would help in a heart beat. Especially since it can leak oil onto the back tire. My buddy bought my 12 so I still work on it since he doesn't have the knowledge. Looking back I should have kept that bike as a stable mate when I bought my VN1600. I miss it a lot.
When I get home I will check and see if I still have the pictures of the connectors when I was going though all this and post them up.
What you can do in the mean time is disconnect the plug in the tail, spray it clean with electrical cleaner and let it air dry. Do this each time prior to going for a ride and coming back. It will prevent the oil from leaking out of the plug and into your tail. The whole stator will eventually need to be replaced due to the wires being saturated with oil already.
Again if you go to the dealer tell them what it is and DON'T LET THEM TRY TO GLUE THE WIRES AT THE PLUG IN THE TAIL. It MUST be done in the case due to that is where the oil originates and once it saturates the wire you will have heat and a possible fire issue.