Worf/Picard know the children are likely in immediate danger at all times. In a very cynical view, they double as shields. Times were more peaceful, but it's dangerous now, not just on Enterprise, but home-worlds too, many families/children see the ships as home.

Wesley is the only child Picard feels comfortable interacting with because he is training to serve in Star Fleet, closer in his mind to a soldier. Wesley understands the danger and commitments through his mother and regarding his father's death. Others Jean-Luc can't help but see as the children they are. And Worf's obsession with turning Alexander into a warrior is really because of the imminent danger.