I have rosemary growing in different parts of my garden, Susan, each with a different microclimate. Yes, mine go dormant in the winter but manage to show new life each spring - except for one planted quite close to the house which surprised me by staying 'alive' all through winter.
I don't get snow here, but I do get very harsh frosts.
If you keep yours indoors (which I've never done) I would say yes, water it. Rain falls outside during my winter. :)