Someone may actually know of a Greek text on winemaking. But there are some technical reasons why this might not have been a huge problem, a few things in favor of the pre-Pasteur brewer and winemaker who couldn't understand the microbial life they were breeding . Yeast first ferment sugar into alcohol. Acetobacter then turn the alcohol ( ethanol) into acetic acid. Acetobacter like relatively high pH, over 5: so, less acidic than the juice of a lot of wine grape varieties. Sourer juice would have tended to make less vinegar. I am not sure if the Greek habit of diluting wine with water was motivated by a wish to have intelligent conversation during the banquet. But if the wine was really sour, diluting it would have also made it easier to drink. Also, yeast fermentation of sugar into alcohol is often quite rapid, a matter of weeks ( depending on the temperature) . The acetobacter fermentation of ethanol to acetic acid takes more time, typically months. If you're willing to drink the wine pretty soon after it's made, vinegar won't be a problem.

They may have also learned to use less bruised and damaged fruit, to chase away fruit flies ( that can transport acetobacter), to clean the fermentation vats and amphora with wood ash or other alkali compounds that would tend to remove bacterial growth. And they may have also simply gone along with whatever happened. Vinegar is handy stuff, and an occasional amphora of it may have commanded a decent price that was not far from the price of wine.

This is common enough information, but Duncan & Acton's 1967 Progressive Winemaking is good reading for anyone wanting to understand the process.