I have a suggestion for something to consider, but I'm no acoustic expert - so I don't know if it will work well. Perhaps cover the opening with a fabric, such as a curtain or other dampening material.
When I had company one time, I blocked one of my archways (for privacy, not acoustics) using a sheet, two shower curtain rings (for mounting), and two 3M removable hooks like these. So, it's something that you could try really cheaply and without any permanent modifications. If it works well and you like it, then you could look at nicer fabric and mounting hardware.