I'm trying to make a silk farm myself so I'm interested to hear how well this works.
My method was to create a seperate stockpile just for cave spiders. As soon as I see a new trap appear in the "cave spider" stockpile, I forbid the spider, but not the trap. This ensures no-one will eat them before i'm ready for step 2.
Step 2 will be to tame the spiders. This won't be possible until the dungeon master shows up, but should be possible given they have the [PET_EXOTIC] tag, am I right? So once I have tame cave spiders, I just wait for dwarves to adopt them. Then i'll have non-biting non-edible non-cat-fodder cave spiders wandering around and (hopefully) spreading webs everywhere.
Step 3, harvest the webs.
Step 4,
Step 5, PROFIT
Seperate stockpile is good. I just did it by queing singluar 'catch live animal' jobs so the anouncment of the job completion would pop up every time my trapper caught something and I could zoom straight to it to forbid spiders and move everything else into a cage to free up the trap again.
I wasn't sure how well tamed spiders would work, which is why I didn't try that. As strays, or indeed as pets, I would guess that even if they do make webs (which I don't know if they do), they would do so in such high traffic areas that the dwarfs would trample them all before you had a chance to collect them.
It seems spiders are still getting into my fortress from the chasm, so they do apparently spawn repeatedly. I've been releasing one at a time into my web room and waiting for it to vanish before releasing the next, but I've still got plenty in stock. I think as far as slik farms go, its opperating well enough. I've probably got enough silk to make clothes for everyone if I wished.