I asked about this on our last cruise because we plan to travel with friends who have major allergies. One of the children is allergic to peanuts.
First, notify
DCL at the time of your reservation. If it is "only" a peanut issue, they are used to this and won't need to bring any special food on board. They are willing to stock special food for a guest and cook it individually if required, but need a 30 day notice.
If it is noted at the time of your reservation, room service and your serving staff should already be aware. However, on the first night of the cruise, your server should ask everyone at your table about any allergies or dietary concerns that they should be aware of. They will then be sure that your orders and what is served are "ok" with the allergy situation.
The biggest concern is that you are "on your own" at the buffets, although the head servers are very knowledgeable and will guide you. There are peanuts at the ice cream station on deck 9.
If needed, they will seat you at a "peanut free" table, where no one at the table will be able to order anything cooked in peanut oil or using any nut products. They generally handle this by seating your family at an individual table and then leaving things up to the mom.
I was really quite satisfied with the explanations I received, and the answers convinced my friend that doing a cruise with her family was possible and safe.