Best way of capturing the Legendary Dogs?

Seeing as they always flee.

1) Get a fast Pokemon with False Swipe. Find the beasts and use False Swipe. Let them run. Repeat until they're at 1 HP.
2) Get a fast Pokemon with a sleep move. Find the beasts and put them to sleep. Let them run.
3) Every time you run into them, they will have 1 HP and be asleep. Toss a Quick Ball or any other powerful PokeBall and hope it works. Repeat until they're caught.

You can use Mean Look to keep them from running, but I honestly find the above method to be easier since you don't actually have to battle them.

What worked for me...I stayed in one spot for a really long time and I eventually encountered both of them. I was on whatever Route it is that wild Snobulls are on and I was searching for it while there wasn't a swarm. I encountered and caught both Raikou and Entei WAY before I even saw a Snobull on that route. The funny thing about it is, I wasn't even looking for them.

For catching them once you've encountered them, I would recommend using someone from the Gengar or Crobat family to use "Mean Look" so they can't run away...but you still have the issue of them "Roaring". Sleep is a good way of dealing with this. Use a pokemon in the high 20s or low 30s to inflict damage on them (anything higher and you may run the risk of unintentionally KOing them). Quick or Fast balls work really well on them too.

Also, you are guaranteed at least 1 attack on them anytime you encounter them.

I bred a Ralts with Mean Look, raised it to the level where it learns Hypnosis, and then TM'd false swipe and taunt onto it. Worked very well with all 3 dogs but the last guy took forever to catch, I kept resetting to check the nature and one of the times I caught him in like 2 balls but every other time was 15-20+ balls. With the first 2 dogs I got em in about 4 duskballs each.

Oh and to finish out that strategy After he learns hypnosis I evo'd him into Gallade and then applied those TMs, And then what you do is Mean Look them so they can't run, then on your next turn Taunt them so they can't roar, then false swipe his ass to 1hp and hypnosis. Every few turns Taunt will run out so make sure to re-taunt on the next turn. Worked without fail for me, they weren't going anywhere. Same for Latias/Latios and everything else I come across.

If you have Quick Balls than use those. A friend of mine caught Entei with one. No battling necessary. I find that Misdrevous or Gengar are really good against them.

From: HungryLuma | #007
Here's a crazy suggestion. Get 6 Wobbuffet's that know Shadow Tag, and put them in your team. (If you didn't know, Shadow Tag prevents pokemon from escaping in a battle.) Knowing that the Legendary Dogs would use Roar to make you bring out another pokemon, your next Wobbuffet that knows Shadow Tag would come out. It'll be impossible for any of the Legendary Dogs to escape.
If you can't do that, then just get a fast pokemon that knows Mean Look.

Roar ends the battle if a wild pokemon uses it.

^No, I used Mean Look with a Gastly on Entei in HG, and it worked fine(Until it Roared it into it's Pokeball)