Who is at fault if my web designer used a photo without permission and now I've been sent a letter demanding $6,000 for the use?

I don't think the issue is who is at fault. The issue is "who is using someone else's copyrighted picture?" Looks like you pal.

Your website your problem. You may have a cause of action against the web designer - but that is after you settle with the photo owner.

If the web designer was working for you (an employee), you are responsible for the violation. If the designer was an independent contractor, either the designer or you both may be liable (depending on what the contract between you says). The degree of liability may also depend on how the photo was used on the site and how much of the original work the photo was part of ended up on your site. Check first to see if the photo is still protected by copyright and make sure that the person claiming ownership actually does own it or have a copyright license to use it.

If an an image has been used without the permission of it's copyright holder on a website, both the designer and the owner of the website carry liability and that liability is independent of each other. Sometimes a designer will take care fo the problem in order to salvage the relationship with their client, but that is not usually the case, from our experience.