This might just be me but, i'd never let movers handle my game collection for a myriad of reasons (theft, damage, etc). Thing is, most people don't think to take any particular care of that sort of stuff because it's just 'old junk' like a DVD movie to them. My personal advice would be to do it yourself.
I just recently moved in January and had to do this myself, though my move was only about 20 minutes away so I was able to do multiple trips. My solution though was to buy these fairly large black totes that are incredibly durable I imagine, I then lined up the totes with bubble wrap and layered the games in them in a non crushing fashion, and kept enough space at the top so that if the lid bowed at all it wouldn't damage anything. I think i was able to fit most of my stuff among maybe 12 totes or so. Still a lot, but in this instance I would always recommend handling it yourself if possible. Things like humidity and temperature can also become an issue depending on what you have (Sega Saturn games seem real bad with this)
Then again, i'm kind of a nervous wreck with my stuff, but I want it to stay nice since I did a lot of work to keep them nice as is.