Why is the area around my child's eyes and nose all red?

Irritation/allergies. Many things cause redness around the nose. Could be irritation from rubbing the nose or eyes constantly due to a cold or allergies. This could also lead to infections such as cellulitis or impetigo. If you are concerned about the redness getting much worse or if it hurts your child, you should see your physician.
Allergy or infection. Could be allergies this time of year but could also be an infection called periorbital cellulitis or its close cousin orbital cellulitis. Periorbital and orbital cellulitis are both potentially fatal so your child needs to be seen asap.