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
Later: what’s up with the obesity rates???
7-up or not, I have watched plenty of parents giving sugar sodas or other calorie laden drinks like sports beverages as a steady diet to young children. I can't recall any that were straight up infants like in this ad. But certainly very young children. And those children, to my eye, did appear obese. Not to mention being set on a path they will struggle to get off of, later in life.
So much modern food is terrible for you, but addictive, and advertised everywhere.
Even juice is awful. And we’ve selected for more and more sugary fruits through selective breeding.