As a parent, I wouldn't have any problem with that, other than the fact that I'd definitely want to taste some of the food!
As a cook, I wouldn't hesitate to serve it to kids. I'm assuming they're old enough to actually like bbq.
If it were loaded with alcohol, that might be another story, esp since most kids don't like anything that actually tastes like alcohol.
The only time I'm careful about cooking with alcohol, even though it's burned off, is if I know I'm serving to a recovering alcoholic. Then I discreetly ask if they'd prefer I leave it out. Or sometimes I just leave it out or make something else.