A Near Miss incident occurred when a joint of 13 3/8 inch casing fell down the V-door and continued off the catwalk, when the single joint casing elevators (SJE) were not latched and pinned correctly. The incident occurred on the 60th joint of casing that was run, using a casing running tool. With the winch line and sling removed from the joint of the casing, the Driller was signaled to pick up the casing. When the weight was taken up on the casing elevators, the elevators opened and the casing fell out and eventually came to rest against the wire line unit off of the catwalk. The consequence of this incident was slight damage to the wire line unit but with potential for a fatality.