Hi Bob,
My immediate reaction would be to question whether you need an Arduino for this. You could just wire the power switch of the winch to a button on the stairs and press the button when you want the door to open.
If you want to use the Arduino, you might want to look at this product: Redirect Notice
Alternatively, you could use the Arduino to power a relay (through a transistor), which could switch on and off a load plugged into the mains.
s.