Large numbers of spiders says you have large numbers of insects for them to prey on. Maybe you can figure out what's causing the insects to thrive in your green house to reduce their food source.
The other possibility that comes to my mind is to encourage predators such as wasps who would feed the spiders to their young. The complication with this, especially in an enclosed space, is that most of the wasps I know hunt spiders can also give painful stings to humans. Do you have the option to welcome lizards and small snakes into your green house?
Like the previous responses, I tend to think of spiders in terms of their food items. We have quite a few but I don't kill them (though I would dispose of a dangerous one).
If you're determined to get rid of spiders: Have you tried using a shop vac with a wand to just do all the edges, corners, etc. in your greenhouse? I don't think anything is going to permanently do away with spiders, but shop vaccing would get lots and would have the added advantage of cleaning things out, probably getting a lot of other bugs, too. Then you could compost all that nice protein.
Diatomaceous earth actually does nothing to spiders. When I first moved into my place I was determined to keep the spiders at bay. They love my entryway and live in all the crevices of my siding. I put massive amounts of DE into each of the crevices and all I accomplished was turning my black spiders white. They were white for MONTHS. It was actually kind of funny to see these sugar-coated looking spiders coming out every night.
Apparently, since they walk with their bodies up off the ground, there's no friction for the DE to cut them up like it does to the bugs that drag themselves around.
I have since come to terms with them. They don't bother me, I don't bother them and they eat some of the mosquitos so that's pretty cool. I will smush them if they try to mess with me though!
I'm not sure there is anything to eat in my greenhouse, as it is newly established. It's rare I see another bug, though there are plenty outside. They may be feeding on microscopic mites and the like. I'm trying EM-1 from Terraganics to improve the soils bacterial or microbial life. I'm not sure if that is feeding them. I'm in an area in NE Arizona that has a lot of Juniper Tree "duff", or composted matter under them, and the spiders are legion (lots and lots).
The DE isn't affecting the ants either (which it is supposed to according to people here and elsewhere). Massive amounts of DE are finally affecting my grubworms in the compost pile (Japanese Beetle kids), as they have been outnumbering, and probably eating, my earthworms.
~ Permaculture is enriching...Farming... is just scratching the surface ~
Are they web spinning spiders, or hunting spiders?
It might be you just had something with a spider egg sack or two inside the green house, and since it is enclosed they couldn't get out and spread out to establish territory of their own.
If there isn't any other bugs, the spiders will start killing eachother.
"Where will you drive your own picket stake? Where will you choose to make your stand? Give me a threshold, a specific point at which you will finally stop running, at which you will finally fight back." (Derrick Jensen)
If it wasn't in your green house, I'd say get some ducks. We used to have so many spiders that when we'd walk in the grass, they would flock away from us and there would be a spider every 1/2 inch for two feet around our step. This was all over our property (1-2 acres were grass. I'm sure they were on our other three acres, but you couldn't se them scurry as easily). They were harmless little grass spiders, but they sure were disconcerting.
Then we got ducks. Now, I step in the grass and there are no more spiders. I just don't see them at all. The ducks probably ate both them and their food. This is about 10 ducks over the span of two acres, over two years.
I wouldn't think you'd want a duck in your green house. They'd soon get bored/hungry and eat your plants unless the plants are all up about 2+ feet off the ground. But, if anyone else has a spider problem outside, ducks just might be your answer... Also, if you have ducks around your green house, you might also greatly reduce your spider population inside the green house. I don't see many spiders in my house any more, either, and the ducks never come in here...