Essentially, any place that you go has the potential for nut-contamination, but the risk is especially high in any place that is frequented by children.

The cause can be as simple as a reaction from your child's touching a railing on a set of stairs. If the railing has been contaminated, your child's hand could pick up the protein. Without proper and immediate hand washing - the next time their hands go into their mouth or onto something that goes into their mouth - they risk an allergic reaction.

By now we all realize that it is impossible to avoid situations where your child is risk of a reaction - basically, they are always at risk. It is how you handle the situation that counts. Control the situation, and be prepared. You cannot stop your child from touching everything he wants to touch - but you can teach him that with thorough hand washing the risk is much less as well as the all important factor of keeping their hands out of their mouth.

When you go to play/entertainment facilities with your allergic child, take your own snacks and drinks unless you have checked ahead to verify they have safe alternatives. Be sure to personalize your child's cup so that they will always recognize it and use it - this will prevent them from using a cup that is identical to theirs but not theirs. Let them write their own name on their cup or draw a sunshine on it for them, this will make it unique to them.

If you need to eat while at the facility and do not have food with you, be sure to ask all of the appropriate questions, or perhaps purchase only pre-packaged foods that you recognize as safe.

One other suggestion I have is to take a baby blanket or some other lightweight cloth to put on the table top where you are going to be eating. As you likely know, cleanliness is not foremost at many of these facilities.

For more info, see The Peanut Allergy Handbook.