A good Balsamic vinegar that has been reduced to a thick, syrupy consistency is a wonderful addition to all sorts of dishes, from jazzing up Asparagus to amazing Brussels Sprouts to an oddly tangy addition to desserts, having a Balsamic reduction on hand is a kitchen staple.
Turn down the heat to the lowest setting, just enough to make the Balsamic vinegar simmer
Stir the Balsamic vinegar occassionally
With the heat still applied, let the Balsamic vinegar reduced by one half of its original volume
Always allow the Balsamic vinegar to cool completely before storing
You will know when the Balsamic reduction is ready because two things happen. First it turns from a tart vinegar flavour to a sweet sugary flavour. Second it becomes syrupy rather than watery. You're generally aiming for a consistency somewhere between pancake syrup and maple syrup.
The Balsamic reduction is reasonably stable if you store it in a tightly sealed container in the refrigerator. Whilst I cannot keep a Balsamic reduction around for very long because it gets used, if you chose a good quality Balsamic vinegar to start with,
If you started off with a Balsamic vinegar that was perhaps less than stellar, and you find after a few weeks in the refrigerator that it has lost some of its "wow" factor, and perhaps is starting to look a little separated and crystallized, you can bring it back to its former glory by gently heating it in a saucepan for 10 or so minutes.