Does Butterbeer contain alcohol?
Alcohol use isn't unheard of in Potterverse. As we know, Hagrid likes to imbibe, and so do Mundungus Fletcher, Professor Slughorn, and Winky the House-elf. In fact, Winky is able to become intoxicated on Butterbeer. J.K. Rowling has always stressed the differences between humans and house-elves, though.
Professor Slughorn breaks out the oak-matured meade for Harry and Ron in Half-Blood Prince. As well, in HBP, Harry is able to keep Hagrid and Slughorn's bottle of wine refilling itself, so apparently there's no underage restriction on the magical production of alcohol in Potterverse.
Everyone in the Order -- adults and teens alike -- at the Burrow has Firewhisky in honor of Mad-Eye Moody in Deathly Hallows.
Butterbeer is also served in The Three Broomsticks. This one might mean nothing, as a person can get a glass of Coke in a Muggle bar or pub just as easily as a mixed drink or a pint. Also, drinking laws are more liberal in Europe and the UK, I believe?
There are myriad Butterbeer recipes online, both alcoholic and non-alcoholic recipes. I also kind of poked around The Wizarding World of Harry Potter and The Three Broomsticks, but no official recipes are posted. Like Disney, Universal does not allow alcohol, so I doubt the theme park is a good resource for confirmation either way (as all their beverages would presumably be alcohol-free).
So does Butterbeer contain alcohol?
I will not make a Butterbeer tag.