Just had our daughter's first haircut here. Our daughter sat in my wife's lap (both having coverings). The barber was very patient and worked carefully. The parents' participation in controlling their child is key. It wasn't a perfect cut, but it's for a 13-month old; not sure of any toddler that can sit still for that long, even with the cartoons and toys to keep them distracted. Not too expensive and they offer simple bang trims for a fraction of the cost of a hair cut (very nice). They even gave a certificate for her first cut.
BTW, I find it hilarious that someone who reviewed this place thinks wet hair will mold... Hair is made of dead skin cells for anyone who doesn't know; it's OK if it is wet in a bag. Your hair will be there long after you're gone.