A cold basement family room can be caused by no cold air return connection from the room to the furnace. Even though the ceiling heat vents are open, the basement is cold and damp when the upstairs is comfortable. To make a closed room comfortable you need to have air in, and air out. Your ceiling heat vent pumps warm air into the room yet it does not have a return to the furnace. (Related: Cool upstairs without air-conditioner cut furnace vent)
Cold damp air is heavier than warm dry air. Therefore, it accumulates in the basement. Drawing the cold, damp, stale air, from the basement floor and sending it through the furnace and back to the whole house will improve the environment in the basement. Installing a cold air vent from the family room to the furnace should provide a constant change of air when the furnace is running.
Adding the vent from the family room to the furnace should be easy. Start the job by cutting a square to round transition in the wall between the family room and the furnace. Locate it so a six-inch flex vent pipe can be run from the transition to the furnace. Then cut a six-inch hole in the cold air return connecting to the furnace. Install a collar in the hole and connect the flex vent from the transition to the collar. Install a grill on the family room side of the transition and your job is complete.
Your family room could be saved for about $25 in parts and two hours of work. This vent saves money in the summer when the air conditioning is on. Cold air (because it is heavy) collects in the basement. Thanks to the new vent, the furnace grabs the cold air and sends it throughout the house.