dnd 5e – What are the DM’s responsibilities when dealing with a character’s death?

This is about Best Practices, not responsibilities.

You ask about responsibilities and there aren’t any, formally.
Informally, the objective of the DM is to facilitate the play experience for the players. There is a practical need to work with the player, once the session is over and before the next session, to get their next character set up and ready to fit into the game world. I’ve done that literally dozens of times, probably hundreds. I have also been the player in that situation dozens of times.

  • Good DMs have always opened up some time to help me get my new PC
    ready for the next session with the group.

Should they leave? Should they just sit quietly and watch? What can
they do? What should I, as DM, do?

It depends on the player.

What I usually do when it comes up during a session

I usually have the dead character’s player, for the rest of that session, do all of the die rolling and movement for the monsters that the party is fighting. I’ve been doing this since I started DMing in the late 1970’s. Very few players don’t enjoy this. Most of them jump right in – when they do it relieves the DM of a lot of detailed work, and (importantly) keeps the player engaged for the rest of the session. (This was very important for pre teens, I found).

For those rare ones who do not embrace that option, I have generally put them to work in creating their next character. They can stay in the room, or, if they don’t want the distraction of the session going on while they do that, move to another room.

For those who get upset and / or rage quit

Yeah, it happens. Some people get very upset about their character’s death. I don’t see it that much anymore, but I saw it a lot among teenaged players and a very few adults.

For those cases:

  1. Express condolences. (I did this even in old school games. I never
    took joy nor pride in the PC dying – some DMs seemed to …)

  2. Remind them that this is a game.

  3. Ask them to make a new character, and encourage them to start right
    now. Sometimes this does not work, and that leaves …

  4. Ask them to come back later, or for the next session, when they have
    cooled down.

Those are DM best practices that I’ve seen, and implemented, over a lot of D&D years.

The Impromptu Funeral – a group option if your players like this

I was in one very memorable group in a high lethality campaign (AD&D 1e) who would, after a battle where a PC died, put together an impromptu funeral pyre and each player would say something nice about the dead PC before the flames burned out – unless hot pursuit was an issue. In a few cases of the latter, we had memorial services at a local tavern during the session wrap up, or at the beginning of the next session.
I (a player) maintained the “Hall of Heroes” notebook where the char sheets of dead PCs was kept. (about a dozen). That small three ring binder is still in a box in my pile of old D&D stuff in the attic.

Proper Prior Planning

DMG p. 236 offers this:

Multiple characters can be a good idea in a game that features nonstop
peril and a high rate of character death. If your group agrees to the
premise, have each player keep one or two additional characters on
hand, ready to jump in whenever the current character dies. Each time
the main character gains a level, the backup characters do as well

If you all are running a campaign with a high lethality rate, or have embraced that tone of a campaign, then having a back up character already put together(one for each player) is a best practice. You still have the matter of what to do ‘for the rest of the session’ or ‘until I can fit in a Meet the New PC situation’ which takes us back to “run the monsters” as a good way to keep the player engaged until the session ends, or the meet up scene arises.

Bonus: this approach is also useful if a player gets tired of a PC and wants to retire (or suicide) the PC and bring in a different one. We’ve had five instances of that in my current shared-world campaign that I DM with my brother.