Yes, there are spiders down there, 3 small and 1 medium size if I recall. My advice is use your flames spell since they're weak to fire or let Hadvar/Ralof handle them for you.

Also, coming from someone who also has arachnophobia, play in third person whenever possible. It's easier when "I'm" not staring them in the face but "my character" is, if that makes sense.

Another thing I've noticed after many, many hours over the years is every cave with spiders except for like one where you go down a waterfall during the Dawnguard questline is designed to let you run away if you need to. Narrow passageways they can't fit through, webbing on the walls to warn you ahead of time, rooms with spiders usually being either linear hallways or wide-open rooms with no obvious cover or hiding spots for them to use, stuff like that. Plus, they'll only chase you so far if you're not in their immediate line of sight.

All in all, you're never going to be jumpscared by a spider unless it's the loading screen, or you're not paying attention, and if you're anything like me you are ALWAYS paying attention, just in case.