The tradition of brewing fermented beverages for consumption is pretty widespread across all human cultures. Root beer, Ginger beer, Ginger Ale, Sodas, are no different. Please note that I will be using beer for ease, the beverages brewed may or may not have contained hops, depending on where the beverage was from.

Root beer, along with all sodas, all come from the tradition of brewing small beer/ale. Small beer used to be brewed for easy consumption by children, or people with health problems. Essentially, you would brew a beer but you would stop the fermentation process far earlier, meaning that you would get a beer with very low alcohol content.

Brewing the beverage was helpful because we are talking about a time before substantial public sanitation. Waterborne diseases were a matter of concern for everyone. Any sort of brewing process for beer (excluding raw ale), involves mashing, which means you'll be boiling the liquid for a good length of time, which means the beverage becomes pasteurized. So, if the beverage was properly made, you can be certain that you won't be giving your kids, or grand-dad, cholera.

Which brings us back to root beer. Before we invented far more handy faster ways of carbonating a liquid, fermentation was the only way to do it. The western pallet had been accustomed to the fermentation. When the Americas were settled, we learned from the natives that Sassafras had certain medicinal qualities. Pharmacists began peddling sassafras syrups, and in rare cases, root beer. At the time, making root beer still meant you had to ferment the liquid, which meant that all root beer had alcohol in it, albeit in small quantities. They were intended to serve as a tonic at the time, but in the late 19th century, you'd start seeing alcohol-free versions make their appearance.

So, to answer your question, a traditional root beer should contain alcohol, in nearly insignificant quantity though. Since they used to be brewed just as you would a beer, there is no getting away from natural fermentation and therefore alcoholic content. If you were to brew a traditional root beer, and not stop the fermentation, you would indeed get a beverage with higher alcoholic content very much resembling a beer.

Sources:

for the origins, Fizz: How Soda Shook Up the World.

for anything related to brewing a root beer.. Well, that's first hand knowledge, I've brewed alcoholic root beer before. There is plenty of easy to find recipes online too.