The first step to catching Latios/Latias is to get the legendary on your Pokedex.
- If you have Sapphire, you are looking for Latias
- If you have Ruby, you are looking for Latios
- If you have Emerald, you choose which Lati you want after beating the game
If you're playing Ruby or Sapphire and you've already beaten your game, you've likely bumped into your Lati at least once. However, if not, getting your Lati registered in the day can be very time consuming. Since the chance of bumping into your Lati in the while randomly is fairly low, I recommend being patient and continuing to play your game normally and traveling around the grass until your Lati appears rather than actively seeking it out.
However, if you have a friend who has the Lati you are looking for, you can easily register the Lati on your dex by having your friend put the Lati in their party and then mixing records with you. If you go to their secret base in your game, you should be able to battle them, and they will have the Lati in their party!
After that, the next step it to find a Pokemon with an ability that stops opponents from running away. There are two abilities in the game that do this: Shadow Tag and Arena Trap. However, Arena Trap does not work on flying or levitating Pokemon, and the only Pokemon with that ability anyways is the unevolved Trapinch.
So that leaves us with two Pokemon who have the Shadow Tag ability.
You can get a Wobbuffet by evolving Wynaut egg you get from the old man in Lavaridge or catch one from the Safari Zone west of Lilycove. You'll want to train your Wobbuffet to about level 60, so it can take your Lati's attacks fairly well. Since Wobbuffet's limited moveset make it a pain to train, I recommend using Experience Share and Rare Candies.
Once you have your Lati Catcher ready, try to meet the legendary. Remember to stock up on a lot of Pokeballs, several Full Restores, and maybe a few X Defends first.
Since your Lati will be level 40, you can also put a Pokemon lower than level 40 first in your party. If you use Repels, your Lati would still be able to appear because it is a higher level than your lead Pokemon. This trick saves time by allowing you to skip extra wild Pokemon battles, but is optional.
Once you're prepared, go to a place where two routes meet. Go back and forth between the two routes and check your Pokedex each time to see if your Lati is on the same route as you. Once you find that you are on the same route, save (in case your Lati faints), then search the grass and water on the route until you bump into the legendary.
Now, you're at the catching. If you don't already have your trapping Pokemon first in your party, switch to it.
If you're using Wobuffest, keep using Mirror Coat until your Lati uses Psychic for the first time. Make sure to take note of approximately how much HP damage the Mirror Coat returned. Then keep using Mirror Coat until the Lati is down to approximately a third of its HP. If you're using another Pokemon, just use a move that does damage and keep track of how much damage you're doing.
Don't lower your Lati's HP too much lower than 1/3 because it is likely to start using Struggle soon.
Once the HP is at a good amount, keep throwing Pokeballs of any type while using healing items on your Pokemon as necessary. If the Lati starts using Struggle, you should use an X Defend or two if you brought any to lessen the recoil damage it does to itself.
If the legendary faints, turn off your game and start going back and forth between routes again. But if not....
Congratulations on your new Latios/Latias!