- Taunt. Since Sleep is typically spread through the use of a status move like Spore, Sleep Powder, Dark Void etc., Taunt is a good way to stop the opponent from putting your Pokemon to sleep. Besides these moves, the only thing worth worrying about is the ability Effect Spore, which is fairly uncommon anyway.
- Grass-type Pokemon. Grass type Pokemon are immune to the powder moves that put Pokemon to sleep (Spore and Sleep Powder), so simply having one on your team gives you a very easy switch in to most methods of putting Pokemon to sleep. Dark Void is only learned by Darkrai and Smeargle, so it's not really worth worrying about.
- Abilities Vital Spirit and Insomnia. Any Pokemon with these is immune to sleep, regardless of how it was caused. The ability Sap Sipper may also be useful as it means you absorb Grass type moves, and Spore and Sleep Powder both happen to be Grass type moves.
You can also use a Pokemon that uses Guts or Toxic Boost as an ability paired with a Toxic Orb or Flame Orb. The orbs will give the Pokemon a status effect that will activate their abilities, and since you can't double up on status moves, they become immune to sleep and any other status problem as an added bonus.
Another method is using Pokemon that can heal your Pokemon from sleep. Pokemon with the Natural Cure ability are good as this, as are "clerics" that use Heal Bell or Aromatherapy to remove status problems like sleep from the team.