The risk of PC death is essential to the game for me. As a player and as a DM. I am unsatisfied when I play in a game in which nobody dies. It doesn't feel like there is any true risk. But, I think different players react differently to PC death. Some members of our group just leave the session when their character dies. Sometimes, they don't come back for a while. There are players who take risks and understand that sometimes adventuring is lethal, and there are overly-cautious players who want to exploit every possible advantage before proceeding. It is a difficult balance to strike.
I think the most important thing to do as a DM is to be fair in applying the rules. Some good advice I read recently in a 20-year old RPG is to give the player a chance, even a very small last chance. If you do that, the player will feel better about the outcome.
Here's a recent example from a D&D game I'm running. PC A died while PC B was trying to carry him away to safety. The combat continued, and the other PCs defeated the foe. After reading the rules for a few minutes, Player A thought that I applied his first stabilization roll too soon. I made him roll immediately because he fell on his turn after an attack of opportunity that he provoked resulted in a critical hit that put him in the negatives. Player A felt that he should have rolled his first stabilization on his next turn (which is what the PHB says) and therefore PC A should have 1 more hit point to his name--putting him at -9 instead of -10 and DEAD. I considered it on the spot and decided it was reasonable to allow PC B a chance to make a Heal check to try to re-stabilize PC A, and failing that to allow PC A one more stabilization roll. It didn't affect the story too much and gave PC A a slim chance of success. I did not allow any of the other PCs to break off combat retroactively to try to help as that would affect the way the story unfolded.
Player B had to roll a 16 or better on PC B's untrained & low-Wisdom Heal skill check to re-stabilize PC A. The d20 fell as Player A nervously prepared his percentile dice to try again to roll 10% or less. Player B made the check, and everyone was happy. I suspect that if both rolls had failed, Player A would have been accepting of the result and just made up a new character.
So, make the threat of PC death a possibility; but give the players every chance you can to avoid it. In the end, make it the fault of their dice if you possbily can. You can't blade a bad roll! (But you can sure blame a bad DM!)