I was at Rock Run ATV Park here in Pennsylvania this year for an organized event. I did see one person who strapped a car seat to the front rack of their atv and had a small child strapped in.
From what I gathered it was a family and they used this set up to travel 2 miles or so to get from the camping area to the events area up a dusty 2 lane road.

That said I have seen multiple low speed stupid things happen where adults were able to jump out of the way and a quad rolled over.
I am talking under 5 mph.

Look at it as risk vs reward
Is the risk of a mishap where there is even a chance of injury worth the smiles and giggles?
I have placed my son 2 1/2 in front of me to take my quad from around back to the front of the house to load it and I had one close call where he reached up and almost got the throttle.
Since that sobering experience I decided it just isnt worth the reward.
****
Short answer is -- as long as there is no law against it he is your child and you are the responsible adult so if you decide to do this then the decision and any outcome is your responsibility.

 

I actually agree with sand hog on this one- I have two boys 20 months old and a 7 year old. The 7 year old has his own atv and will also ride on my 2-up around the local trails sometimes- but I move real slow and the trails are flat and open.
I'd never even consider bolting, strapping, welding, taping or glueing a toddler seat to the racks of an atv for a trail ride! Not because I don't believe you can't control your atv- but even I have had unexpected spontaneous control issues while riding otherwise safe trails. A lose or broken tierod can jam your steering and make you lose control very easy- and hit a tree. I've been riding for 25 years and have had at least a dozen tie-rods fail on my atv's- it's not uncommon. You'd never forgive yourself if something bad happened to your toddler, who was strapped to an out of control atv and got hurt badly- bad bad bad idea period!

I also agree with the use of a neck pad for children 5-12 on their own atv in trail conditions.

My son got his first 100cc chinese atv at 5 years old- those chinese atv's tip pretty easy and he rolled it a few times. They are only about 80-90 lbs and close to the ground, and dont move very fast, but it was a good lesson for him.

I also on a regular basis pull the kids around on a 4'x8' wagon / hay ride that they love. I take them through the trail at about 5mph.-

 

What if those people aren't too bright- what makes them the authority on what's safe for your child? Is it worth the risk to a toddler? It's a bit difficult to answer your question without injecting some opinion, after all, you did post it on a public board- and they are only opinions.