This is actually quite a popular blog post topic. And different experts will tell you different things.
Feel free to search the internet and you will come across multiple photographers' take on this. They will advise you to consider experience, style, education, facilities, etc. etc.
Personally, I believe the choice of newborn and baby photographer often ends up being quite an emotional one, rather than the result of ticking a checklist of photographer qualities and capabilites.
But hey, don't take my word for it :)
The below quotes, by great photographers, may prove useful in helping you choose who is the right photographer for you!
“If the photographer is interested in the people in front of his lens, and if he is compassionate, it’s already a lot. The instrument is not the camera but the photographer.”
— Eve Arnold
“Great photography is about depth of feeling, not depth of field.”
— Peter Adams
“It is more important to click with people than to click the shutter.”
— Alfred Eisenstaedt