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.
The answer is a combination of physics and physiology.
The warm water in the shower very quickly heats up the air in the shower, and warms up your skin. It also drives up the humidity of the air in the shower. You acclimate very quickly to the temperature/humidity conditions in the shower as being "normal".
With the door left open a crack, you allow convection to occur... the warmer, more humid air in the shower escapes out the space at the top, replaced with cooler, drier air from the bathroom. Convection is driven by the difference in density between the warm humid air and the cooler drier air; it's not just the difference in temperature which alters density... a molecule of water (H2O) is lighter than a molecule of either of the principle components of air (N2 and O2), so humidity lowers density for equal temperature and pressure.
The draft feels unusually cool because it is somewhat cooler than what you've acclimated to (stronger subjective experience), and because the drier air in the draft is able to draw much more heat from your skin through evaporation than the warmer, more humid air.
Best Answer
It depends on what do you mean when you say effective. You are absolutely right when you say that the only way for your oven to cool is by diffusing and radiating its heat to your house, regardless of whether its door is open or not. So the total heat energy transferred from your oven to your room is fixed. But when the door is open, the power, or energy per unit time, is larger.
The word "efficiency" is usually used to denote the amount of energy produced divided by the amount invested, and is not appropriate in this context.