Unfortunately, I don't think a non-pesticide route will be effective enough in this case. You probably should do a fumigation or something to get rid of the existing ones. Maybe once a month if it persists. I don't think you'll ever be rid of the problem, as the trees are probably the source for the spiders, though they can come from the ground as well.
Keeping your windows closed will help a bit and is probably the best defense, the daytime heat and sun can sometimes get rid of them.
Typically they will probably just be trying to make a web at night, and shouldn't jump out at you. But, I know if I see one crawling that I'll scurry to get rid of it.