As far as I am seeing on Snopes and CBS News, this 7-Up ad was a real ad in the 1950's.
NOTE! There is almost the same image circulating that claims to be about giving beer to infants. That is a FAKE ad. It looks virtually identical as this one, but replaces 7-Up with beer. Same baby etc.
There is also a "Soda Pop Board of America" fake ad, with the tagline, "For a Better Start in Life, Start Cola Earlier". It also features a photo of a baby, but looks different from this. That ad is also FAKE -> Hoax Ad
My grandmother's go-to cure for nausea was 7-up and orange juice. Any grandkid that wound up staying home from school got it. She swore it had to be 7-up, even though the stuff had the same ingredients as any other lemon-lime soda. Gotta wonder how much this kind of marketing played into that insistence.
I'm just glad it was OJ and not milk.
FWIW, it works okay as a nausea remedy, but my theory is that this is just because it's got fluids, calories, and is usually palatable for someone with mild nausea.