1. Personality: To me, this is important, if I am going to spend the big bucks for a trainer and be subjected to exercises that I wouldn't attempt on my own because they hurt, I would hope this person would have a good personality to help me get through the pain.
At my gym, you meet with the Training Manager and that person helps identify the type of trainer that you need and then goes and schedules you with that person. You don’t get to meet your PT unless they happen to be working at that time and are free to chat; which is unlikely if your trainer is a good one. (Most of the popular/good trainers at a gym are always with a client unless there is a cancellation... therefore, they wouldn't be able to speak to you.) Make sure that you can identify the type of trainer you are looking for (bootcamp style, drill sergent, fun, high energy, motivational, challenging, ect).
2. Time Conscious: It is important that your trainer is time conscious so that you can get the best session possible. The gym is not the doctor’s office where it is acceptable to always be running late. You want your trainer to arrive on time and use the full hour that you have paid for. If your trainer is running late, or is late because of his/her previous client, you should expect that your trainer will still give you the full hour even if this means that you will run into their next client’s schedule.
3. Planning: Not all of my trainers have done this, but my favorite one did, so I wanted to mention this. The trainer should have a plan for your session based on your goals. Some PTs may actually have this written (which is AMAZING!), while others will have it in their head, but if you ask about the plan, they should be able to relay the details to you with little to no difficulty. This means that they have really looked into your goals and taken the time to develop a strategy for you to meet them.
How would you judge if a trainer is good?