Pay attention to how much they ask you about your business. Your web designer should really want to get to know you and your product or service that you are selling. That way they can deliver a new website product that will engage your customers and sell your service well.
Look at the other websites they have designed and see if they match your aesthetic taste. A great designer should be able to create exactly what you are looking for but it's a sure bet if their past web design projects really appeal to you.
Are they graphic designers as well or just developers? It's important that you hire one company to do both the graphics and the coding. Otherwise, things can get lost in translation as you are trying to communicate to your developer what you want which they intern have to communicate to the designer. Ideally, you want to speak directly with the designer who is creating your website design and graphics.
Ask them about internet marketing and search engine optimization. Many web design firms are clueless about SEO. This can be a major issue if they are replacing a website that already ranks well in Google. If they do not follow certain SEO protocols when launching your new site it could completely tank your rankings permanently and cost you a lot of money in lost business.