It sucks when you loose a good character, typically you did something dumb and have to pay the price. Of course death in D&D is a very subjective thing . . . what am I saying? I'd tell the player to cowboy up and roll another character, then I'd kill that one too.
But how do you cope with an emotional player? Do you think...hey he/she might have come to the table with emotional baggage that will ruin the game for everyone?
Somebody actually having a melt down over a death? Well, if they get violent or belligerent, they are gone. We are all friends at the table, and folks should be mature enough to lose gracefully. If they blame me, as a DM, I always make sure that I had nothing to do with it. In my games, one dies from dumb decisions, or because they rolled the dice and a risk didn't pan out. If this isn't the case, and I am DMing some killer dungeon, like Tomb of Horrors, then I will let everybody know prior to play that the adventure is going to be a brutal one and they may not want to risk characters that they are afraid to lose. If a player gets really sad or upset in a none violent way, I may pull them aside and talk to them privately about what is going on, and try and help them the best I can, but the thing is that my games can get viscous, I had the benefit of a DM who allowed me to fail when I was starting, and it made me a better player, so I continue that tradition. I usually suggest that a new player not play a full character right away, and keep it simple. You don't want to spend 3 hours working on a Character Sheet just to have the dude die in the first session because you think that you are a super hero at first level, it takes time to acquire the skills to keep them alive. If you make death common, the sting isn't as painful when a character does bite it. After a bit you just laugh while everybody ribs you and you roll up a new one.