I know this thread is dead but I figure I'd get it current for all of those using "the google."
If you want to ship live crab, you really have two options:
1) Small, moist, insulated, and cold.
2) Large, aerated, liquid, and cold.
The second is more for commercial users or those who want to hold crab indefinitely. I will tell you right now that my company makes systems to do this. It's not something you want to do yourself. You can check my site http://www.ari1.com/products
But the first is fairly easy. What you have to remember is that crab take in oxygen and food, then dump out waste, largely in the form of ammonia and and carbon dioxide. Both are lethal to crab if they build up. When the crab are "purged," they will create less waste and ship much better. If you leave them in a trap with no food, they will purge in 24 hours. You want them to be free of mud, purged, and cold.
Now that your crab are purged (you can also ask the person you buy them from), you want to get their body temperature very close to 32 degrees F. Too much below that and they'll die. Much above that and they'll die. DO NOT MESS WITH THE CRAB. They hate that! The more they respirate, the more they metabolize and the closer they are to dying.
So get a foam box, ice cubes in zip locs, and a clean, wet towel (some people use seaweed, which is actually better to use). The most crucial part to remember is that you DO NOT want the crab to be submerged in ANY depth of water. As soon as they are, they'll excrete ammonia and CO2 and, you guessed it, they'll die.
Put a damp towel on the bottom of the cooler, place the crab in the cooler, then another damp towel on top of them, then the ice packs. NO DRY ICE! Dry ice is CO2. Any guesses what CO2 will do? Yep. Kill em.
Now ship immediately. Label the box live seafood and ship them as quick as humanly possible! Good luck.