Yes,it does.Certain stages can work as an advantage or disadvantage for certain characters to fight one.For example,Little Mac will have a fine time playing on stages like Boxing Ring and Final Destination,but others like Gaur Plain,Wooly World and 75m(practicaly,this stage is already unplayable outside of 8-player smash) just won't work for the character.Moving stages are the ones like Orbital Gate Assault and Mushroomy Kingdom(I consider this one to be unplayable as well) who make the biggest difference on that topic,and it will depend whether your character can fight and move on at the same time with the stage.However,because most tournaments like to stick around to simple stages like Battlefield,Final Destination,Town and City and Smashville,players won't bother too much on that aspect,but it will be worth remindering when some decide to play tournaments like this every once in a while.