I'm a new DM, and the majority of my players (including myself) are either completely new to D&D (or any tabletop RPG for that matter) or very inexperienced.
(By the way we're playing D&D 5e, so I'll make a few references to it but I didn't tag this question D&D 5e since I'm open to answers in general, not just related to D&D 5e).
We've been playing weekly for a few months now, and overall things are going great. The players are all great people and we're having a lot of fun.
But one hitch at the moment is that one of the players is scared of his character dying to the point where he has a ton of real-life stress about it. This is despite the fact that he's the only character that has never dropped to 0 HP and had to make death saving throws. So I'd say his fear has less to do with the reality of in-game danger, and is more just a personal issue. He's said that he is scared of putting all this time into developing his character week after week only to have the character die.
As a result of this fear, I worked with the player to develop some fun RP story for his character. One of the Unearthed Arcana subclasses for D&D 5e is the Shadow Sorcerer, who has a class ability that can prevent death. So he wanted to multiclass into that, and from a story perspective it's very interesting – embracing powers of shadow and evil in order to fend off death. Classic!
So I made a very special session the last time we played where he and the other players ventured into the Shadowfell in a very creepy, eerie session that I told him going into the session would result in him taking his first level in Shadow Sorcerer.
And as part of the whole thematic notion of this class having an ability to withstand death, part of the session involved bringing him down to near death before allowing these powers to awaken, which allowed them to escape the Shadowfell. I feel like this all worked very well, and the other players had a blast.
But, the player in question didn't have a lot of fun because he was so scared of dying. I thought that might be the case before the session started and even contemplated telling him that he couldn't die in this session, just to ease his mind. But I thought that would eliminate the dramatic tension of the session.
So I feel bad for this player, but I'm not quite sure what to do to help him not be so stressed out. Again, I'm already not going hard on him (to reiterate, he's never even dropped to 0 HP yet) so I don't think it's a matter of easing up on the difficulty. It's just the general fear that his character could die.
So, what can I do to ease the stress of this player, either in-game or out-of-game, and improve his enjoyment by not being so scared of character death?
Make resurrection a clear, viable option
If the player is purely afraid of losing his character, enable him to hold on to him, at some cost, even if he is to die. While this may cheapen the narrative and/or the suspense of the game, your players' enjoyment of the game is arguably more important than narrative cohesion. Consider a cleric in town, which the party could solicit healing services from, a guild or faction which the player may be a part of, which could provide some assistance in these matters, or any party members which may possess the necessary spells.