It's kind of a Catch 22.
If STI wants to see the game before they take and wrap it, then you won't have time to pack it securely. STI looks it over for 1 second, puts moving blankets over the game, then wraps it in stretch wrap. That's their "professional packing".
You can do a MUCH better job packing yourself, but then STI might argue that they never saw the condition of the game - so take lots of pictures!
I leave it up to the client if they want STI looking at the game and STI wrapping it, or better packing with me wrapping it.
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If there is anyplace likely to get damaged, it the corners of the game.
I steal these heavy duty fiberboard corner protectors from work, and run them along all the edges of the game:
http://www.uline.com/BL_8402/Heavy-Duty-Edge-Protectors
L_shaped_cardboard_corner_cardboard_corners_protective.jpg
I like these to go on top of the L protectors for the 8 corners of the backbox:
carton-corner-protectors_PDT00196.jpg
Plastic banding is good to keep all the L protectors in place before you stretch wrap. It comes in a kit with a tensioning tool and locking clips:
http://www.uline.com/BL_2752/Uline-Poly-Strapping-Kits
HD_2752_T05.jpg
Put the clips on the bottom, so if the game gets banged around, the clips won't imprint themselves into the visible parts of the cab.