What is the best way to deal with cave spiders?

Lava. It will kill the spiders, destroy the webs and spawners too, and potentially make it bright enough to stop them being hostile.

Usually, whenever I encounter a mob spawner, I note where the block is, put several blocks around it (to make sure mobs can't come out), tunnel my way underground to where the mob spawner is, and whack at it until the spawner is destroyed.

That way, I'm directly underneath the mob spawner, I'm in little danger, and I can just hack away at it until it's destroyed.

TNT

Now that you can't click-to-ignite, I find a quick 1-2 with TNT/redstone torch works well as an alternative. This will kill the spiders, and destroy the spawners. It'll also make a cool noise!