As mentioned in an earlier installment, it is forbidden for a person to pray, recite Berachot, or learn Torah if there is even a tiny amount of bodily waste on his body. For this reason, a person should clean himself with water, or at least moistened paper, after moving his bowls, to ensure that no traces of excrement remain on the body.
Young children, quite obviously, cannot be expected to maintain this strict standard of cleanliness. Nevertheless, it is permissible for parents to learn Torah, pray and recite Berachot with their young child, even if it can be assumed that the child had not thoroughly cleaned himself after using the restroom. As long as there is no smell of excrement emanating from the child's body, this is permissible in the interest of Hinuch (the child's religious education).
However, once a child reaches the age of 7 or 8, where he can be taught to clean himself thoroughly, with water, parents must ensure that he does so, as maintaining proper hygiene is part of a child's training in Halachic observance.