The primary reasons I think taking children to Everest Base Camp is a bad idea:

The first reason is that the sun will fry your skin quickly at that altitude.

the 2nd reason is that the sun is really hard on a person's eyes to be at that altitude

The 3rd reason is that if one or both of their parents dies on climbing Mt. Everest who is going to care for the children? If you study how many people get caught in Bad weather on Mt. Everest where the weather can be good and then quickly become not survivable for a variety of reasons, death is very common there, especially among people who aren't properly prepared either physically, mentally or equipment wise. 

And then even if you are perfectly prepared all sorts of things can go wrong unexpectedly and people get ill or injured and then they cannot come down off the mountain without help or at all. And only the lightest of people could Sherpas carry off the mountain. It is unlikely that they could bring a 200 to 300 pound man who is over 6 feet 5 inches off the mountain for example.

Although in the right circumstances with some kind of sled to take a body down they might recover someone alive or their body.

This is why taking children to Everest Base Camp doesn't seem like a useful or good idea because parents and children might not fully or completely survive it. However, I have heard of helicopters landing there to pick people up at Base Camp if people can afford it. I'm thinking 20,000 to 50,000 dollars for that sort of thing in a rescue by Helicopter. But even then the weather has to be perfect and not whited out or foggy or cloudy or windy to do this in reality and not theory.