Yes balsamic vinegar does go bad eventually but balsamic vinegar should last at least 3 years and sometimes longer.
The shelf life of balsamic vinegar is around 3 years and after 3 years it starts losing it's freshness although it's still okay to use the balsamic vinegar past that time.
Although for the best results you should use the balsamic vinegar before 3 years.
Just make sure when you open the bottle of balsamic vinegar to keep the lid on it when not in use and I would also recommend refrigerating it as well even though you're not required to do so.
Unopened and completely sealed bottles of balsamic vinegar should last 10 years or even longer but mostly I use things such as vinegar and balsamic vinegar well before 10 years so it's usually not a problem.