Re: Is it wise to keep a door open to cellar to let cold air fall away?

In principle your idea should work.  Cold air being denser than warm air 
it should move downwards and in the process displace the air that is below 
it.  But having a single door open may not solve your problem because the 
cold air may not all move through the door that is open.  The use of fans 
or some other circulation system might achive this.

You could of course experiment.  Make sure the thermostats are set for the 
different areas of the house and take temperature readings in the 
different areas with the basement door open and not open.

My gut reaction is to say that the basement door should remain closed.  
The air in the basement is probably the coldest in your house and so 
denser than air in the floor above the basement.  For that reason there 
should be no movement of air into the basement.  But if your basement is 
warmer than the first floor, then opening the door might prove useful.

What you really want is convection to help you move warm air up and cold 
air down.  

A final factor to consider is that by keeping the basement door closed you 
reduce the volume of air that you have to heat and so the upper areas of 
the house are heated faster and with less expenditure of energy.  There is 
probably more to be gained financially in this way and is the reason that 
the basement door has been kept closed for all these years.  

Hope this helps you.