1) Visit the websites of a few design firms and look at their work and client list. Look at their about page. 

Based on your goals for design — what do you want the design to do for your business or org — select 2 or 3.

2) Contact those 2 – 3 firms to schedule an initial consultation. Most designers and firms will not charge for an initial meeting. Be prepared to share your mission, current situation, and goals honestly.

3) Have the meeting or phone call. What you want to do hear is assess if you like the people enough to consider working with them. Can you trust them to carry your vision forward? Do they understand your situation and what the design needs to do?

Do they talk in business language (things like risk, demographics, psychographics, ROI) or design language (things like balance, gestalt, color)?

Do they understand who they're designing for (it's not really for you, it's for your customers)?

Good designers and design firms are not order takers. They're strategists and they understand that design solves business problems. So you want to work with those who get that.

4) Do you like the people you talked with? Did they pick up your ball and carry it forward a step or two? Did they open up your thinking in a way that not just impresses you but that helps you see greater possibilities? How do you feel at the end of the meeting or the call?

5) Can they do it within your budget? Or, can you allot more money to the project?

So essentially it comes down to trust and likability, as with any relationship.

It's similar from my side as a designer. I don't work with everyone who contacts me about a project. I want to work with people I like and trust. It's a relationship.