I don't drink, and neither does my wife, so I don't exactly spend much time in bars - so I haven't seen this, but that's not really evidence of anything in my book. I also don't know the law in this regard, so if it's actually illegal, obviously you shouldn't do it.
Additionally, as was pointed out in other answers, the effect of alcohol on a nursing baby is not insignificant; but I think this isn't really asking about that (and it's no different if you drink at home).
I would point out that (either legally or socially) disallowing women with infants from bars is a subtle form of gender discrimination. You're doing something that has very little impact on men, and has a significant impact on women, between the single mothers and the fact that (even in this day and age) women tend to be the caretakers of infants. Barring women from bars if they have an infant in a wrap (or stroller) limits their ability to go out and socialize, and that's going to have a disparate impact on women versus men.
Certainly it's possible to hire a babysitter or a cousin or (...) to watch your baby, and I'd think that is a better option; but requiring it, either legally or socially, would mean many single women and mothers who have less helpful husbands would have to choose between going out and spending $50-$100 on babysitting, or not socializing with their friends; thus, poor and middle class single women would often have to forgo socialization. It's not as bad as being banned from breastfeeding in public places, but it's in the same direction.
Obviously the actual risks of having a baby in the bar would need to be weighed against that disparate impact, but I suspect the actual risks are relatively low, as long as either she isn't breastfeeding, doesn't drink herself, or is careful to time things so she won't have a high alcohol content in her breastmilk at feeding time.
The latter seems difficult, but in the former two cases, I don't really see a practical reason for forbidding it. The reason minors aren't allowed in bars is the concern that it will have an impact on their moral development (ie, they'll be exposed to curse words, fighting, etc., drinking culture, lewd women, etc.), which won't really effect a four month old. As far as mouthing things, well, that's a concern anywhere: I don't think bars are especially concerning compared to restaurants or movie theaters. As long as the baby is kept close to the mother and she doesn't drink too much, I think the baby would be perfectly fine.