Does open stairs between basement and main living cost extra electricity bill?
It depends if your &quot;heating bill&quot; already includes heating the basement.  If you have forced air heating or cooling with the basement on the same zone as the living area, it should make no difference because the air is already circulating and mixing.
If you heat your basement and it's on a separate zone, your bill might go up as the zones will not work as efficiently.  If you do not currently heat your basement, your bill for the upstairs might go up. Or, it might go down if your basement is naturally warm (say, that's where the furnace is) and the new arrangement allows warm air to rise upstairs more efficiently.
You didn't ask about cooling but if you have cooling upstairs but not downstairs and your basement is not naturally cool in summer, your cooling bill will go up, possibly a lot as the cool air generated for upstairs will have to fill the basement before it can begin cooling the living area.