Cold air under base board heat from floor

You have baseboard heat AND forced air? Because if you HAD baseboard heat at one point and someone CHANGED to forced air, but just left the old baseboards there, I would consider removing them, then taking the hardwood all the way to the base of the wall.

If you are still using the baseboard heaters, then you have to be concerned for the air flow coming into them, which is likely why someone left the hardwood out of that space. In other words filling that gap may result in restricting the air flow into the bottom of the baseboard heaters, reducing their effectiveness. You would have to then try to find out where the draft is actually coming from and fix it there.