Gingers in this Northerly climate are tricky things. I manage to get my Daniel Weeks ginger to bloom late in the fall, just before the frost hits. I also have a H. gardnerianum (from a root bought from the Kona airport in Hawaii several years back) which just sent up an inflorescence this fall, but isn't quite getting to bloom fully. It doesn't bloom as early as the Daniel Weeks I've discovered. I'm starting to wonder if there are certain plants that just are meant to be grown south of z6 and not worth the effort North of that. I love zone denial and all, but its a lot of effort for little enjoyment.
To answer your question:
What I do is dig them up before the frost does major damage. I put them in as large a pot as I can handle, splitting off any outliers and roots that can't make it into the pot. Sometimes I completely spray off all topsoil and replant with fresh soiless potting mix to reduce bugs and diseases. From there they live in my screened porch for couple weeks to finish blossoming and then they live either in the house or more often in the basement by a south window. Give them just a little water to keep them on life support for the duration of the winter/early spring months. Too much water and they tend to rot. They don't need to be completely happy and look a bit rough by spring, but they come back vigorously once they are planted in the spring after the last frost.
Alternatively you can prune off the tops, hose off the soil if you like and store them dry in peat for the winter as you would with cannas. They are solid rhizomes should survive.
Hope this helps!