Small Spiders will quickly run toward Albert West on sight, and bite him rapidly. Like all enemies, their attacks slow West a bit when they hit, so even a small group of two or three spiders can lock down West if he gets hit. Shielding the hits or dodging back are almost necessary to escape if West is caught, especially since gunshots may go over them if they are close enough to hit West. If the associated brain bug loses sight of West, the small spiders will likely retreat back to it. If the associated brain bug is killed, the small spiders will die a few moments after.
Interestingly, small spiders don't seem to have much of a will on their own, instead only attacking or retreating from West when the brain bug decides. This can be observed when a brain bug becomes a cocoon: the small spiders may be left behind and will never attack West, since they don't have the brain bug to direct them anymore. This can be exploited by closing doors on the brain bug, which can be easy when alone with one since the difference in movement speed between the small spiders and the brain bug is huge. Once the door is closed and they are separated, the spiders will keep trying to retreat back to the brain bug.
Small spiders only attack with melee, so retaliating with your own melee will suffice, and is actually recommended since it can be difficult to hit them with guns. If there are too many spiders around, a photon grenade can easily clear them out. If they are spread out sufficiently, a lucky shotgun shot can clear them, but the power required may not be worth it.