It has a normal winter dormacy. Let it get really dry between waterings.

Here is a link that might be useful: Zingiber

"Red Cone" ginger is not the normal term used for Zingiber zerumbet. "Red Cone" usually refers to Alpinia purpurata. Zinbiger zerumbet is referred to as "Red Pinecone" Ginger.

You need to specify which one you have, because the care is much different between the 2.

Alpinia purpurata "Red Cone" ginger is an evergreen Tropical zone 10-11 plant that may not do well overwinter under a growlight. It may do better in a very sunny window, and should not be allowed to go much below 55 degrees F to keep it from stalling out. It has no natural dormancy period and generally blooms on 2 year old or older growth, terminally. If you have this one, you should water it maybe once a week while it is in your house this winter and keep it warm and well lit.

Zingiber zerumbet "Red Pinecone Ginger" or "Shampoo Ginger" is a deciduous ginger with a natural winter dormancy period. It blooms basally ( directly from the ground) and is starting to brown and go dormant all over the FL peninsula at the moment. If you have this one, you should stop watering it now.

Thanks Bihai
I looked it up and my ginger is REDCONE GINGER PLANT - ALPINIA PURPURATA JUNGLE KING. It really isn't growing good at all now. I may have lost it, but there is still a bit of green on the stem. I will move it to a sunny window and hope it improves. Any suggestions to saving this baby is appreciated.