Could I use spiders to catch unwanted insects?
What you describe is called biological pest control. Lacewings and earwigs are more common, but there are ways to use spiders.

http://unossc1.undp.org/GSSDAcademy/SIE/Docs/Vol4/Rearing_spiders.pdf

http://www.annualreviews.org/doi/abs/10.1146/annurev.en.29.010184.001503?journalCode=ento