I'd be lying if I said I didn't have an irrational fear of spiders, too. Come to think of it, I don't like many kinds of insects, either. Cockroaches gross me out.
Regarding the spiders specfically: they must be my least favorite kind of enemy to fight. It doesn't help that the devs made them as big as they possibly could and still have the mob fit on a single tile. At least they didn't make them burst into a gooey mess of green slime when they die. (ew)
It also doesn't help that at one point, the devs decided it would be fun if a secret wall opened and some spiders started wandering around to likely attack you from the rear at a bad moment. Nothing quite like turning around and seeing a spider there when you weren't expecting it.
To the OP: I know this isn't really a fix, but what I do is I pretend they aren't really spiders. I pretend they are robotic spiders, or some kind of magical creature that just looks like a spider. I know it can be tough to re-program your own brain but it works for me. Maybe try doing this, telling yourself in your head, "These aren't really spiders. They aren't really giant 300 lb. spiders that have hairs on their many legs that are so big that I can see them waving at me."
If that doesn't work, I'm afraid you will have to simply wait for some kind soul to produce the no-spider mod.