You believe you have a problem that he can solve, and you are willing to pay him for his time, knowledge, equipment, ingenuity, experience, and courage. The first step is to educate yourself. At the very least, understand how trees should be pruned. Beyond that, it will pay you to learn how trees work, so you can be sure that all work causes the least possible damage, and works with the tree's natural defenses. Read through our information, read the findings of tree scientists such as Dr. Alex Shigo and Dr. Kim Coder and others .
Experience is hard to measure.
There is a world of difference between 20 years of experience and one year repeated 20 times. One way to estimate an arborist's experience is to look at his work on other trees and talk with their owners, keeping in mind what you have learned about tree biology. Salesmanship is part of business, but in tree work anything beyond what is needed is likely to be harmful. Make a list of what really needs to be done, and get comparative prices on that work only. Be open-minded, but not gullible.