Hi Debbie We had the same problem....a nicely finished basement, wonderful in the summer, but just too cold to use in the winter. I solved this by installing a natural gas, ventless, wall mounted heater I picked up from my local home improvement warehouse (Hechingers) for about $150 or so. The difference is night and day. It is now a warm and cozy area, where my wife and I spend our evenings reading, listening to music and working our in our exercise room! The hard part for us was getting the gas piped over to where we wanted to mount the unit. It took some doing, but it is doable. If you don't have experience plumbing gas, get a plumber or gas fitter.
The open fireplace is without doubt the worst form of space heating available. It acts as a large heat pump, pumping lots and lots of warm inside air to the outdoors. This air is replaced by nice, fresh, cold outside air through the small cracks and crevices around doors, windows, flues, etc, etc. The net effect is that open fireplaces work to cool your house....a kind of winter air conditioner!