Generally speaking, our "Best By" dates are suggestions and not "Expiration" dates. There is never a food safety risk – it's simply a flavor and aroma precaution.
Unlike some perishable foods, balsamic vinegar typically gets better with age! Its flavor may change slightly over time, becoming more complex and potentially more intense. However, the natural sugars in the balsamic can "crystalize" over long periods of time. As always, trust your senses - if it smells or tastes off, it's best to discard it.