Their role in Christian art should perhaps not be surprising
Spiders have always caught the imagination of the world in a web of curiosity. However, throughout history their hard exoskeleton, thin dangly legs and juicy eating habits have given them a more sinister reputation.
The Greco-Roman world viewed spiders as a symbol of pride. This was due to a myth surrounding Arachne the weaver and Athena. Arachne, boasting of her skill as a weaver, challenged the goddess to a weaving contest. When Arachne won, Athena in her rage turned Arachne into a spider. It is from this episode that we get the word arachnid.
In the medieval period monks used spiders to represent evil. For example, spiders were often seen in connection to the miser, who bleeds the poor in a similar way to how the spider consumes the fluid out of its catch.
Spiders were also more generally seen as a representation of the Devil. A spider is one who sets a trap for its prey with its web and so too does the Devil, setting traps of temptation, always “seeking some one to devour” (1 Peter 5:8).
So the spider, even though we can now see their natural purpose more clearly, were seen in the ancient world as evil, diabolical and to be avoided.