Dad here. I simply don’t offer if it involves their kids, it is far too likely to be misconstrued. Plus, that mom is probably doing all she can to keep it together, and knowing that someone noticed could put her in a worse place mentally. If I’m out with my wife, she has offered out help before, and that is much less likely to be judged. It’s just a fact.

(I will offer to do things like take the shopping cart back, but I would do that regardless of whether they have kids with them or not.)

I have sometimes seen little kids wander off while their mom wasn’t looking. I do pay attention to where they go, and try to keep them in sight. Sometimes it might have only looked like they were wandering, only to find they were going toward another trusted adult. But other times you can see that moment of panic on the mom’s face when they realize what happened. It’s enough to be able to point and say “they went that way” and they can go get the kid themselves.

I won’t interfere unless the kid is going into a dangerous situation, like a street or into water. Even then, I won’t touch the kid unless it’s the last possible thing to do to keep them safe, I am more likely to get in their way and ask them where their parents are.

Dad here, I have both been the guy, and been the parent and in a safe area where you have some control basic human trust can be enough. I always ask/offer, try not to be creepy.

I often start off, with something like "I can sympathize, I have two myself, would you like some help? I would say 50% of the time they say yes, and the rest its a combo of fear or embarrassment because they think needing help is shameful, which it is not, your community exists to help and thats how humans did it for 100k’s years until the 1950’s ruined everything.