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).

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.

Try to get on top of the spawner, and break it from there. The spiders won't get to you with no vertical wall to climb.

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!

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.

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.

Minecraft Wiki has a table of damage done by weapons.

Swords: The best ordinary weapon (by damage numbers) you can readily make is an iron sword, and the second-best is a stone sword, which is very easy to craft since it only requires wood and cobblestone.

Bow and arrows: I strongly recommend keeping a bow and arrows, if you can, because it enables you to kill monsters at a far distance (especially important since spiders jump at you and creepers will explode without actually touching you). A bow requires three string and three sticks, so the hard part is killing enough spiders to get the string. You could:

•Kill spiders in the morning, when they will only attack you after you attack them and so won't swarm. •Build a trap for wandering spiders; for example, a wall of cactus which you're on the other side of. However, this isn't as easy now that spiders can climb; you'll have to put an overhang on top so that they climb up it and stop. •Or just kill spiders with your stone or iron sword. Arrows for a bow are made from flint, feathers, and sticks, or are dropped by skeletons. Flint is readily obtained by mining gravel (any gravel that doesn't produce flint can be re-placed and mined again for another chance at flint), and feathers may be obtained easily from chickens.

Burning: It is possible to set enemies on fire, but it doesn't kill them very fast. A more practical possibility is to drop a bucket of lava. The safety of this depends strongly on the terrain (you can outrun lava unless you're clumsy, but the real hazard is if the flow blocks you from walking back to pick up the source), but if you can do it then it will bog down the enemies that try to chase you across the lava and kill them reasonably quickly.

Armor: Also, find a herd of cows and start breeding and slaughtering them to make leather armor (24 leather makes a full set). Armor makes you take a lot less damage, if nothing else so you can get in some aiming and timing practice in those beginning fights. Also, killing cows and chickens will help you learn how far your weapons reach and how to aim at a mob.

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.

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:

well they aren't that hard to kill just have iron armor and a shield ( of course a pickaxe) at least an stone sword and im done. Even if your difficulty is hard