No, the wine in Bible times contained enough alcohol to cause people to get drunk, which would have been more than “a little” alcohol.
The answer is we don’t know the alcohol percentage for a typical ancient wine, but we do know it was strong enough to make people drunk. Therefore we can be confident that the references to Jesus and wine were not in reference to grape juice or wine with very little alcohol.
Any brewer can tell you that it is hard to ferment something “just a little”. Once the yeast gets started, the fermentation will keep going unless you intentionally intervene with chemistry – which was not a topic of understanding in ancient times. To this end, many believe the wine from ancient times was actually stronger in alcohol content than what we are used to today. Therefore they would have added water to the wine to make it drinkable, and avoid drunkenness from a small quantity of drink. Even given this fact, drunkenness was a common topic in both the Old and New Testament, therefore leading us to conclude that the alcohol content was not what we would consider “negligable”. The percentage probably varied, but it still was strong enough to cause drunkenness when not consumed responsibly.
A similar argument is made that references to wine and Jesus were actually references to grape juice (not wine). Another FAQ addresses this question.