A zero-death run is where one tries to accomplish completion of one of the Pikmin games without losing any Pikmin whatsoever. This can be very difficult, even for skilled players. However, there are a number of ways whereby this can be made easier, listed below.
Even if only one of your Pikmin dies, do not blow a hole in your wall, just press the "RESET" button on your console, Select "Reset" from the HOME MENU (Wii), or press the "POWER" button so it goes on and off. This is especially useful in caves, where your progress is saved between sublevels, meaning you will lose little progress.
Holding X+B+Start will also reset the game.
Take your time
Especially in the first game, Pikmin, where enemies can be a lot harder to beat due to a greater lag and several glitches, like the Crushing Glitch, you need to take your time so as to avoid even the slightest of accidents; carefully avoid the enemies' attacks, and if you fail you can always reset the game. Patience is vital when fighting difficult enemies like the Beady Long Legs and Emperor Bulblax, or the likes of the Titan Dweevil in Pikmin 2.
Some enemies are hard to beat without a large onslaught, so take the strongest forces of Pikmin you can muster, and use them correctly. In Pikmin 2, you should always try and defeat enemies with Purple Pikmin whenever possible, because it can be done with relative ease and takes less time. In the first game, there are also a number of alternative, safer ways to defeat enemies, such as using Bomb Rocks. If you throw pikmin while rotating the C-Stick, pikmin can be thrown much faster, saving you more time and deaths. Also, whenever an enemy starts to shake pikmin off, immediately start throwing more, as they will not reach the enemy until the animation is over (mostly).
While the enemies that lurk on the main paths should always be dispatched; many enemies near parts and treasures that are on the road less traveled don't always have to be killed. Most enemies will follow a captain with or without Pikmin, so use a captain to distract a group of enemies while a small group of Pikmin slip by with a part or treasure. In caves one should find the most direct route to the treasures and only kill the enemies on that route. In Pikmin 2, try to use Ultra Bitter Sprays when possible.
Do not overburden yourself
Although it may make the next few days a little easier, killing all the enemies in the level or trying to kill multiple bosses in one day will likely result in multiple resets and frustration.
The Captains punch can take out most of the threatening non-boss enemies rather efficiently. Just be sure to avoid taking too much damage (although this applies less to the first game). Captains can run around the sublevels of later caves to activate any traps and assess the dangers within the sublevel.
Know your enemies
Sometimes, attempting a Zero Death Run is not a good idea during your first shot at Pikmin, but do not senselessly sacrifice your Pikmin; rather, study what the enemies and hazards of the Pikmin games do and look for exploits, easier ways to dodge their attacks and defeat them. Many of the guides here can help you in learning about the enemies, but only by playing the game can you gain experience.
Note: The AI Co-Captain will attack alongside you, but only if he is right next to you as you are attacking.