Crab goes bad very quickly after it dies unless you clean them thoroughly while they are still alive (like within hours). I have eaten cleaned refrigerated crab after 2 days of being dead and not cooked (I cooked them after 2 days) The key indicator is a smell of ammonia. If you get that smell, don't bother. I am a crabber, I just got off the water. I am crabbing 5 days a week on the side :0 Cooked crab lasts about 5 days refrigerated.
The duration in which cooked crab can be kept in the fridge especially if it is fresh from the water is 3 to 5 days. It is advisable that the crab be refrigerated two hours after it has been cooked for it to be well preserved. Visit http://www.stilltasty.com/fooditems/index/16983 for more information.