Lava.
It will kill the spiders, destroy the webs and spawners too, and potentially make it bright enough to stop them being hostile.
gaming.stackexchange.com/users/2198/Alan B
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.
gaming.stackexchange.com/users/11946/Dante
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!
gaming.stackexchange.com/users/3610/fredley
My strategy is to block off the spawn area, so that the spiders can't reach you.
Make a one block hole through which you can see, and mine the spawner, but the spiders can't get through so easily
(cave spiders are smaller than normal spiders, can can fit through a 1x1 hole, but you can easily knock them back if they try to come through).
gaming.stackexchange.com/users/3610/fredley
Surround the entire thing in a 5x5x5 dimension cut-out room with the spawner smack in the middle, freestanding in the air.
Then, make a water current towards the end of one side of the box, light the other 3 sides up, place metal bars, dig a hole underneath, and there's your mob farm.
gaming.stackexchange.com/users/15710/Jacob
When I come across a cave spider spawner I usually surround the area with torches.
(taken from the wiki)
A monster spawner spawns monsters in an 8x8x3 area of the spawner block so place your torches accordingly.
In several cases this means they can spawn through walls in another tunnel.
Also keep in mind that you can disable the spawner (with torches or destroying it) and still have stray spiders hiding high up in corners waiting for you to walk by.
Use your ears to hear them and hunt them down.
gaming.stackexchange.com/users/8506/Dycker
I just fought a large army of cave spiders in survival hard.
It was a big mix of using a water bucket to redirect them and destroy the webs, swinging my sword, and then going and blocking myself off to regenerate, and occasionally use the bucket of water and a fishing rod to collect food.
Took an hour to disable two spawners.
I could have done better, but at least I survived.
gaming.stackexchange.com/users/125654/Iisthellama
Flames or lava cannot destroy mob spawners, but you can place them around the spawner to prevent mobs from spawning or at least make them catch on fire.
The lava will create enough light so that nothing can appear there.
You can also use explosives such as TNT to blow the spawner up, but there's a risk of dying, so I wouldn't recommend to do that.
It could work though.
If you want to keep the spawner to farm the cave spiders, you could create a trap like this one:
gaming.stackexchange.com/users/125954/ShadowWolf
The cave spider spawner isn't too hard to destroy, it can be done in a few seconds using a pickaxe. But it is fairly difficult when cave spiders are still spawning. And it seems that placing torches by/on the spawner doesn't stop them spawning.  Does anyone have any strategies for dealing with this?
What is the best way to deal with cave spiders?
gaming.stackexchange.com
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gaming.stackexchange.com/questions/30756
gaming.stackexchange.com/users/11098/DanielGibbs
Lava will kill the spiders and webs. You can also put blocks around the mob spawner so the spiders can't reach you and tunnel underground to destroy it. Lava, TNT and torches, blocking the spawn area are the most prominent solutions.
Lava could be a solution. Blocking off the area seems to be a valid strategy. TNT and torches could be used to deal with the cave spiders.