I would probably just put a rolled-up towel in the corner junction between the windshield and the dashboard to sop up whatever condensation/frost drips down.
The cover on the outside would probably help some, though not entirely cure the problem. If it were insulated it would help more, perhaps by interposing a blanket or something between the cover and the windshield if that is practical. The basic cause of the condensation, of course, is that the temperature of the inside surface of the glass is colder than the dew point of the air in the cabin (and, if it's frosting rather than dripping, also below freezing). Dehumidifying the air will lower the dew point, but going too far in that direction makes for uncomfortably dry air for you to live in.