All of these things are easier to evaluate when you have more experience, so since you are new to riding - check out a bunch of places so you have a basis for comparison. Ask lots of questions. How do you decide how to pair horses and riders for lessons? What is your cancellation policy? Can I watch a lesson or two? What is a typical progression of a rider riding X times a week? They should also ask you about your previous experience and goals.
I look for indications of safety - are riders wearing helmets? Are people working with horses on the ground wearing helmets? If they have available helmets for riders, are they relatively new (there should be a tag with mfct date inside) and do they recommend you get your own. We were having a discussion about how different two instructors can be regarding wearing helmets for new riders. Instructor A says, “I know in the release you can waive the requirement to wear a helmet because you are over 18, but I have been around a long time and I and all my students wear helmets when mounted.” Instructor B says, “We have a helmet you can borrow if you want, but you don’t have to wear it if you don’t want to.”
Also are the helmets fitted correctly - covering the forehead, not perched on the back of the head?
Barn tidiness is a part of safety - are items laying around in walkways, or stored out of the way? Do the fences and stalls look sturdy and in good repair? Are hoses coiled and hung or laying around? Are there piles of manure all over?
If it is a boarding barn, are owner and emergency numbers posted on the stalls?
Look at the water - is it clean and fresh and in ample supply - keeping in mind if it is an automatic waterer it may appear as a fairly small bowl of water.
Are horses clean or covered in dirt, mud, stains and filth? Do they appear skinny, or a healthy weight?
Do they have barn rules - either posted or on paper? Do they appear to be followed?
Communication skills are great, but don’t always correlate to horsemanship. Only you can decide if not being able to get a hold of your instructor will be a deal breaker.
And always keep in mind - the right barn for you now, may not be the right barn for you some time down the road. I have switched barns due to moves, life changes, changing goals for the barn/trainer, switching disciplines - life happens.