I've had eczema my entire life. So have my children.
There's a diaper cream ( which I can't remember the name of off hand ) that seems to help. You have to ask the pharmacist for it.
It's in a white tube with green printing and it's basically a combo of calamine lotion and zinc. The paste inside is light pink.
I used this on my last baby and it worked great.
It's called something like Calmoseptine I believe.
I hate steroid creams and I can't imagine using it on a baby. Doctors love the stuff, even though all it does is treat symptoms, and not well IMO.
I'd try something like Aveeno Baby eczema cream.
Dove bar soap for sensitive skin, Stelatopia Cream Cleanser, Cetaphil Gentle Skin Cleanser, and CeraVe© Hydrating Cleanser are examples of cleansers often recommended by dermatologists to treat eczema. Cleansers should only be used to wash the diaper area and areas that appear dirty. They do not need to be used on a daily basis.