The best way to take care of eczema is to keep the skin moist as much as possible. Some of the useful tips are: 1. After bath, gently pat dry your baby to retain moisture and immediately apply some moisturizer (Baby Aquaphor is good) to trap the moisture in the skin. 2. Avoid synthetic clothing as it causes more itching in the affected area. 3. Applying thin layer of hydrocortisone cream is helpful. 4. Also aloe vera juice and gel are effective in treating eczema.
My nephew had sever eczema that cleared up when he stopped eating/drinking dairy. He is allergic to the Casein protein in milk. That tends to cause eczema in infants. You might want to look into that! Otherwise, 1% hydrocortizone cream works very well as Riyuj mentions above. I'd probably check with a doctor before using it on a new born!