cut it down cover with mulch, leaves or straw and it will be just fine come next spring

I am also on Long Island. If your basjoo is only 1 foot tall after an entire growing season, something is probably wrong with your planting site or cultivation practices. That plant should be well over 5 feet under normal conditions and even bigger with lots of water and fertilizer.

A one foot plant may not have stored enough energy to make it through the winter.

Do you have it in mostly shade?