Like my other post about encounter balance, this article takes Old-School Essentials for reference, but can be applied to most retro-clones.
The standard rules of most OSR systems simply have characters die at 0 hp.
Many players and GMs feel that's too harsh, especially if you actually roll your class hit die to determine hit points at level 1, meaning you may very well start with 1 or 2 hit points.
That's as bad as it sounds, one hit and you're down... and yet it's very much in line with one failed save and you're dead, which is a constant threat (poison snakes, death spell, pietrification, and so on) even when you have enough hp to take a dozen arrows.
Adhering 100% to those rules is much less harsh if you use Retainers and/or each player plays two or three characters. After all, Old-School Essentials explicitly states it's designed for 6-8 characters, not 3-4. This is the best solution the book has to offer, and if you want to stay within the boundaries of the rules and experience the original gameplay, I suggest you try this route first. This probably is, by the way, where the DCC RPG four-characters-per-player funnel adventures originate from.
But most players prefer to play one character. I get it.
In my DMing career, I've often tweaked death and dying one way or another. Here's a few systems I've tried, plus some I've read around and liked, or not liked, presented to you in small modular bits.
Maximum HP at level 1. That's probably the most common house rule! It has a large impact on character survivability for level 1-3, then evens out at higher levels. Simple, zero book-keeping involved, nothing to remember while playing.
"Roll the body" Save. When you reach 0 hp, you drop down and you might be dead. You make a final Save versus Death when (if) someone checks on you. You pass it, you're back on your feet with 1 hp and still have a chance to make it back from the dungeon. This type of rule can be found, for example, in Dungeon Crawl Classics. If you're left there, you're dead, eaten by monsters or just bled out. This is highly dramatic for sure! Allows for glorious TPKs where no one is there to check on fallen heroes, and works as a high tension final parachute. Because of Saves progression, higher level characters have better and better chances of avoiding Death's door at the very last minute. Starting characters, not so much. Also, some classes definitely have an advantage with this. I'm looking at you, dwarves. Anyway! This requires no book-keeping. The only thing you have to remember is "hey please check on me!" when you go down. A simpler variation is just immediately Save versus Death to, well, Cheat Death as soon as you hit 0 hp.
Negative HPs. As soon as your HPs reach zero or lower, you're "dying", or "bleeding out". You're out of combat, and might die unless helped. Negative HPs have been around since forever, often together with max hp at level 1. Lamentations of the Flame Princess has them. The big difference in implementation is what happens when you're out of HPs. In most cases, you have a maximum amount of negative HPs, which might be fixed (-10, for example, or half your Constitution score). Actual death can happen as a Save versus Death every round to avoid bleeding out until someone tends to your wounds. Nasty GMs might ask you to apply your negative HPs as a modifier to your Save roll, which scales well with higher level characters and their improved saves. Sometimes bleeding out is translated as suffering 1 hp of damage every round, until you hit -10 or another threshold, which means you're finally dead.
I completely, irrationally hate negative HPs. Anyway. You have a little extra book-keeping. Saving every round (or suffering -1 hp per round until -10) works as a clock for other players, forcing them to decide if they want to spend a round to try and save your life somehow.
Or, you know, just leave those HPs the way they are, and try something different if you're determined to increase survivability. Like...
Shields Shall Be Splintered. One of the most popular OSR house rules of all times. You sacrifice your shield and completely avoid damage from one blow. This simple rule originated here in 2008 and has been developed and built upon in many variations, also involving helms and other pieces of gear. If you use this rule, you must decide what happens to magic shields... and probably enforce a "can't carry more than one shield" rule. This rule leaves non-shield-using classes the way they are, though.
A possible variant: anything you are holding in your hands may save you from one blow and is destroyed in the process, but you must succeed in a roll (a Save versus Death, of course) to pull such a feat. Sounds silly? Think how many swords +1, wands, and holy symbols are going to get broken...
Injury/Death and Dismemberment Tables. Another OSR staple. When at 0 hp, instead of dropping down dead, you roll on a table and see what's happened. Sometimes you lose an eye or ear, some time a finger, toe, or limb. Some time you suffer no effect at all, or lose a point of Constitution or other score, and some time you're crushed to bits, adieu. Conceptually, this adds layers of intermediate results to the binary outcome of th Roll the Body/Cheat Death final Save. This post lists so many examples you may check to pick your favorite table or to create your own. Using such tables increases survivability, and may lead to character retirement! Which can be a nice outcome, for a change. Players must keep in mind that losing their sword arm was the alternative to losing their character altogether...
Also keep in mind that you can decide to have injuries and lost stat points to be permanent or temporary. Temporary effects may be recovered in a given amount of time (weeks for example), while the permanent ones may still be recovered with some magic effect: custom spells, rituals, or the obvious Wish solution.