[Wolf spiders are members of the family Lycosidae, from the Ancient Greek word "λύκος" meaning "wolf". They are robust and agile hunters with excellent eyesight They live mostly in solitude and hunt alone, and do not spin webs.] (they do not bite people, who are too big to be considered food.)

phobia reduction is often addressed by small, incremental, steps. first you go to the front door,, see: no spiders; then into the living room, see: no spiders...
add a spider catcher to the mix... this will keep the spider away... use one of those 'dream catchers', maybe ?

it's an irrational fear which many people have, and can be helped by seemingly irrational protections... i wish you well.

you might try adding pictures of spiders to the process, especially the eyes, which are the most alarming part of spiders to many arachnophobes. only hold the picture as close as the 'child' is comfortable with, and gradually bring it closer and closer, maybe even isolating and enlarging the eyes or feet, whichever is most alarming. knowledge is power, and understanding the pieces can make the 'whole' more acceptable.

and, honestly, it is not all that irrational to think that a deadly danger lurking in the house makes it uninhabitable. you might get a spider in a jar and tell your child that you have rid the house of it. and that it is not a threat, but is like a vacuum cleaner, keeping the house free of other bugs and flies. (rational explanations rarely work.)