I have lived with Eczema most of my life. My skin is more sensitive than most babies skin. I know others in the same situation. We each have a certain laundry detergent that works. Mine is All Free & Clear. I would recommend not using any dryer sheets because it leaves a residue on the clothes. However, if you found a dryer sheet that works then it you may be able to use it. There's a chance your child could develop an allergy to it so I would monitor that.
It turns out that I had several skin sensitivities that turned into allergies because I exposed my skin to certain products repeatedly. With my daughter, I'm staying away from any products that are scented and trying to pick products that have as few ingredients as possible. I'm also staying away from "all-natural" products because they tend to have higher concentrations of certain ingredients like essential oils that can create allergies. For example, we use an unscented castile soap for her baths.
Most of my allergies developed when I turned 30. I can now only use basis soap, head & shoulders shampoo and a special toothpaste or baking soda to brush my teeth because I'm allergic to all other products. I can't use Neosporin or get the flu shot because I'm allergic to those. I could go on but I think you get the picture. It's a pain and I don't want my daughter to go through what I'm going through. That's why I'm being picky about products that touch her skin.
That being said, my daughter doesn't have eczema so she may not have any of my issues and your son may outgrow his eczema and not have the allergies I do. I just wanted to make you aware of my extreme situation and what I'm doing.
I have never heard of the banana issue you're having. My nephew has been diagnosed with some food allergies and my SIL needs to be careful with his diet. She's hoping he outgrows this when he turns one like his sister did.
Good luck!
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I wanted to add a few more comments based on what others have said. Vaseline is OK to use. There are lots of rumors about what it is made of and how bad it is but it's actually not bad.
Be careful about using anything with botanicals or essential oils because they can make sensitive skin break out even more. I practically break out thinking about the stuff ;-) A dear friend of mine made the mistake of buying an all natural eczema cure right before her twin daughters first birthday and they both broke out in a severe rash.