How hot does the upstairs need to be before it will heat the basement through convection? During the winter we keep the basement door closed so that the cold air from upstairs won't flow into the basement.  Then we heat the basement and the upstairs gets heated partially by conduction through the floor.  But during the summer, the upstairs is heated while the basement (unfinished) is around 62 degrees F (16 degrees C).  
How hot does it need to be upstairs before opening the basement door will actually heat the basement?  I know hot air rises, but it seems that if there were a massive heat difference between upstairs and downstairs that air turbulence or diffusion would carry some of the heat.  There's an 8' 7" height difference between upstairs and downstairs.
 A: Opening the basement door does not really help to heat the basement - the hotter the air gets upstairs, the larger the difference in density gets, compared to the air in the basement.  
So, convection will not work (- even with turbulences, slight wind - the hot air needs to move some meters down, so that all would not help).
Conduction through the floor will get stronger/more effective with higher temperature difference - so that would help heating at least a little.
Theoretically, opening the door could allow some heating of the basement by radiation from upstairs, if the geometry is suitable.
That would not have a strong effect, of course. 

In total, opening the door does not do much, and the heating of the basement is mainly by conduction.
An interesting aspect is that the increased temperature difference will result in a stronger separation of hot and cold air.
That may reduce heat flow by diffusion - not sure. 

From a practical perspective, it may be relevant that the air in the basement is heated from the ceiling, so much of the heat will end up in a layer of warmer air there.
