How do I clean my utensils after eating to protect someone with a peanut allergy?

Dishwashing. Standard dishwashing should be fine for just about everyone, as long as the dishes are "clean like your mom told you" (no residual food stuck on them). The problem tends to be more when a utensil is used in one dish and then another in the same meal or during food preparation, without washing in between.
Good washing. I agree with answer of dr mcghee, a thorough washing with hot water and soap and making sure no residual food is left behind should be adequate.