Let me start out by saying that Aviation Stack Exchange is not a substitute for medical advice. Most of us aren't doctors (though I have medical training I am certainly not a doctor), and even if we were doctors we are not your doctor and we don't know your medical history (or your child's medical history).
If you're going to be flying with your kids in an unpressurized aircraft consult your pediatrician for their recommendation on supplemental oxygen. There are a number of factors at play, and your doctor will be able to consider all of them and give you comprehensive advice.
That said, let's talk about oxygen in general terms.
There's a lot of good information on supplemental oxygen use out there: AOPA talks about it here, and this website has a very detailed explanation, written by a doctor which get into what's going on as altitude increases (and the partial pressure of oxygen in your lungs decreases).
By regulation you are required to provide supplemental oxygen:
- to the flight crew if you're above 12,500 feet for more than 30 minutes.
- to the flight crew if you're above 14,000 feet for ANY length of time.
- to everyone onboard if you're above 15,000 feet for ANY length of time.
Using the handy chart in the second website that equates to oxygen saturations (SpO2) of roughly 88%, 82%, and 76%, assuming you're breathing ambient air for an extended period of time.
To put this into perspective if you come to the hospital with an oxygen saturation below 90% you will almost universally be put on supplemental oxygen, and most guidelines call for maintaining SpO2 above 92%. (In aviation terms, the partial pressure of oxygen at 10,000 feet usually results in SpO2 around 90%).
So with that information the question now becomes "What is your standard of care for your passengers?" - the FAA sets minimum requirements in the regulations, and they are mainly focused on making sure the flight crew does not become impaired (they are less concerned about your passengers getting loopy, as long as the pilot is OK), but you as a pilot are free to set higher standards based on your own understanding of physiology and aeromedical factors.
I can't tell you what you should do, but I can tell you what I do: On any flight above 10,000 feet I carry supplemental oxygen, I use it, and I provide it to my passengers.
I also carry a pulse oximeter in the aircraft to check oxygen saturation (SpO2) on myself or my passengers as needed (and I set the oxygen flow rate based on having the worst person in the aircraft above 90%).