In choosing a photographer, you would look at his/her works first. Same with choosing a videographer: look at his/her works first. Now, the question is: how to tell if the video is of good quality or not? As any form of art, subjectivity is a given, so you hire a videographer only if you like his/her works. So what's left for me to tell you are the things to pay attention to so that you don't accidentally hire a lousy videographer. I am only going to focus on a couple of very simple things here:
First, pay attention to his/her equipments. Equipments are not everything, of course. But if a professional videographer is unwilling to invest in professional grade equipments, it shows a lack of commitments or aptitude. Since most internal mics don't usually produced good sound quality, are external mics (attached to the camcorder) used? Does he or she shoot with one or more camcorders? Are remote mics used for stage performance involving tap dancing? Etc.
Secondly, true professional videographers rarely shoot videos in fully automatic mode. Just think about it: if it's that easy, why do we need a professional to do that? The fact is that camcorders are not as smart as human and don't know what is important and what is not to the viewer and therefore true professionals rarely trust the camcorder to make all the decisions. But still some "professionals" do it anyway and I was surprised how often they could get away with it. Now let me tell you a simple trick to spot that if this is an indoor performance video, which is what vast majority of my clients have to deal with.
Here is the trick: A good video always have to zoom in and zoom out from time to time, just pay attention when the video zooms out from a mostly stage view to a view that includes the audiences. If people on stage becomes much brighter whenever the zoom-out happens, you can almost be certain that the video was shot in fully automatic mode. When people on stage are over exposed, the details are lost, and it becomes painful to watch. The reason is very simple. Because the audiences sit in the dark, the camcorder, unknowing the audiences are uninteresting part of the frame, will try to compensate for the fact that the audiences are "too dark" and will let more light in, therefore resulting in people on stage being over exposed.
Worse than that, some videos are over exposed through out. It should be even easier to spot that.