There's a great Jerry Cooper lecture on sex in Mesopotamia, where he points out that a carving of a man drinking and having sex at the same time was a standard sign, for hundreds of years. Brothels selling booze seems to be a very old, and obviously good, business decision.

Follow up - would it have been common to have a brothel at Roman bars?

No, not really, at least not in an actual brothel. Roman brothels were quite small--the largest one known to us is a lupanar at Pompeii with only 10 rooms, all of them bedrooms. The vast majority of Roman brothels were small, one or two room affairs--Pompeii is littered with these little structures, over the doorways of which the price of the prostitute's services is listed (on average two asses, according to the New Pauly). These bedrooms were also pretty simple, the ones at Pompeii pretty much consisting of a small brick bed with a mattress on top, and occasionally with erotic paintings on the walls--they're also very small, and the ones at Pompeii are tiny and terribly lit. That said, as the New Pauly notes prostitution didn't happen only at brothels. Streetwalkers were of course common, and Ulpian mentions innkeepers hiring prostitutes and operating them at the inns, although it seems from what he says that this was not a normal or particularly tolerated practice (he says that the prostitutes are hired sub praetextu instrumenti cauponii "under the pretext of inkeeper's work"--that is, pretending that they're servers or maids or whatever exactly it is that you'd hire a woman to work at an inn normally for). Still, it seems to be a fairly common perception in Latin literature that innkeepers are associated with pimps.

But we have to make a distinction between what we would think of by a "bar" or even a pub with the realities of Roman taverns and inns. Latin has several words for different types of public eateries, and in general there wasn't really such a thing as a bar in the modern sense, where you can get just drinks but not really much else--your corner Irish pub does that, sure, but Moe's Tavern is not the same sort of establishment. The Roman caupona, usually translated as "inn," or "tavern," was a place where first and foremost you spent the night. They had a reputation of being seedy places where pimps hung out, particularly the ones in the city (Pompeii's are surrounded by brothels in the nearby side streets)--a person like Cicero would stay with his hospites or at various deversoria that he might own, rather than at an inn. Tabernae and popinae, the closest things to what we might consider a bar, were very different. Tabernae varied in size and quality, but they were generally places you could sit down and enjoy a hot meal and a drink in relative comfort--this was an establishment intended to serve entire meals, rather than simply drinks. One could in theory go into a taberna to simply "grab a drink," but...that wasn't really normal, you got a meal along with your wine. A popina is basically the same thing, but more like a snack shop where you could pick up a quick bite--it's important to realize that popina is also used metaphorically to refer to the food (never the drink) served at these establishments. Consider, for example, that the separate "bar" of restaurants or pubs that is so familiar to us as a counter behind which the alcohol and little else is kept, would not exist in any establishment in the Roman world--tabernae had these funky countertops. What you're looking at resembles the form of a modern bar, but is not. Those are really tables, behind which the tavern-keeper could stand, and those funky little pits in the counter are for heating pots--these things are for meals, not to grab a beer (or in the Roman world, wine). And while tabernae and popinae might be surrounded by brothels (although popinae could usually be found in just about every area of a city, even in the wealthy districts), they didn't really have them, unlike an inn. It was perfectly possible, even convenient, to find and sollicit a prostitute there, but they're not brothels--you'd go somewhere else, probably her brothel, to do your business. After all, a taberna is an eatery, it's not a place where you stay and therefore it has no bedrooms or other accommodations (to say nothing of the owner kicking you the fuck out for messing with his business)