As you've seen from the responses, there are different views on this subject. Here are mine:
Non-swimmers -- Regardless of age, must wear a life jacket (pfd) at all times while on deck, dock, dinghy.
-- 12 and under, at all times when underway (per USCG rules), on deck, dock, dinghy
-- Over 12, as conditions and course warrant
Harnesses -- Subject to exceptions noted below,
- Off-shore, used at all times (day/night, fair/foul weather) when on deck
- In-shore/coastal, as conditions and crew contingent dictate
- Harnesses will be worn in-shore/coastal for overnight passages
- Harnesses will be worn in-shore/coastal when water temps present increased risk of hypothermia
- Harnesses will be used in-shore/coastal when solo- or effectively solo-sailing
- Harnesses may be omitted when parents conclude risk to young children from using is greater than benefit
So, we would always have any non-swimmer wear a pfd. Wearing a harness is not a substitute. Case closed.
Harnesses have their place too. But, harnesses can be tricky with some young kids, particularly toddlers. They only work if the individual is clipped in -- yet kids can unclip in a flash. And as another member pointed out, just like at home/on land, there can be a very serious risk of strangulation when toddlers have unsupervised access to chords, lines, etc.
Sometimes that risk is of the accidental kind, other times it is deliberate on the part of the toddler. But it is very real and you only need to go look at the statistics. Toddlers and chords are not a good combination.
Ultimately, there is no substitute for vigilance. Having young kids aboard does add stress for the parents, and neither pfds nor harnesses are a cure-all. But once kids become competent swimmers, and they are at least able to maintain their heads above water while wearing the pfd, the margin of safety does improve. So early swimming lessons are important too.
We've always favored the Mustang pfds for our kids. But that Crewsaver pfd to which Docksaver linked is a very nice unit.