I'm assuming your gingham is made of cotton or some sort of cotton blend and not silk or a wool blend. You can't put water on a silk or wool blend.
More than likely, if you remove the cushion or flip over the chair there will be a tag that will tell you the materials used in the chair. If you can type the materials out here, I can help you a bit better and suggest a specific cleaner type. If the item says "Dry clean only" you can't use liquids on it, so you'll have to ignore the advice below. Otherwise, you may be good to go.
For now, I'll give general advice.
Test an inconspicuous spot on the chair to make sure water doesn't ruin the color or the fabric.
You can clean up an upholstered (there you go, you can spell it now!) chair two ways, either with a steam cleaner (best way) or by hand.
Kroger and other grocery stores rent out steam cleaners that can be used for carpets, sofas, chairs, and whatever else. It uses water and a mild cleaner, but you can use mild soap or baking soda and vinegar. This method will deep clean and remove stains and make your furniture (or carpet) look like new. Renting a steam cleaner is $20. I love it.
The other way to go is doing it by hand. First, vacuum the chair with the attachment on your vacuum to get rid of any crumbs and loose dirt and dust.
You can hand-clean the chair with a gentle, non-scouring sponge (use a soft sponge or nubby washcloth) and either mild soap, Woolite for babies (it shouldn't be harmful to your child), Dr. Bronners' liquid soap, or a solution of baking soda and water. Make a watery mixture in a bowl, bucket, or spray bottle so you don't have too many bubbles. You can use a nylon bristled brush (try the spa section, car section, dishwashing section, or even one of those nail brushes) to gently scrub any deep stains. Don't grind the stain in, but brush the stain in one motion and in one direction. For darker stains and food stains, you can try baking soda, but if that doesn't work, you can try OxiClean powders. Once the reaction has been made with OxiClean and water, it's not toxic for your child.
For deep down stinky smells, you can put the OxiClean and water solution or baking soda and water solution in a squirt bottle and saturate the area so that it goes through the foam or polyfill. Press down with a fresh, dry towel to sop up any extra water. Make sure there is no accumulated water in the padding (this where the steam cleaner is advantageous, because it automatically sucks up the water).
Let the chair dry for a few days before anybody uses it. Vacuum one last time after it's dry.