How well do they know their industry?
Web design is in constant flux. This means constant design and search changes that need to be kept up with. When changes affect ranking they need to be considered while designing, developing AND maintaining successful websites. This can include anything that makes up your online presence from tag or link structure, citation weight, GMB listings, and mobile practices. Make sure your web designer isn't only good at one thing. Perhaps they do well in ranking your site but your site is still ugly and doesn't convert well. Or perhaps you have a beautiful site that has been sitting on page 3 or years. Ask your designer basic design questions like what "above the fold content" is, or what are the best resources to educate yourself on SEO. Also ask them what their strategy on your site will be? What long-tail keywords your website should focus on, or how they get in-bound links to your site. And check to see where their own site appears in the search results.