Remember that the Earth is a sphere, like a giant ball: so there is no "up" or "down", since a sphere is symmetric. That is, it looks the same no matter what way you look at it. So, people in Australia have just as much right to call themselves "up" as people in the Northern Hemisphere do!
But more to the point, the force that keeps everything on the Earth from falling off it is gravity: it's the pull exerted by one object on another because of its mass. For objects like the Earth, the force exerted by the Earth's mass pulls objects in the direction of the Earth's centre. This means that wherever you are on the Earth, the force is always "down" into the ground. That's what keeps everything on Earth "right-side up", even those in Australia!
Jagadheep adds to above:
Up and Down are all a local perspective. For a person in the northern hemisphere, the zenith position in the southern hemisphere seems to be "down". But for a person in the southern hemisphere, zenith is what he/she will call "up". Gravity is always aligned towards the center of the Earth, and so the direction perpendicular to the surface of Earth (at any location on Earth) is what will be referred to as "up" (locally). Hence, everywhere on Earth, a person is still walking "normally" straight up. An analogy is to refer to directions using "right" and "left. Depending on what cardinal direction you are facing, the terms "right" and "left" will differ. What is "right" to one person will be "left" for another (if he were facing the opposite direction). Thus, "right" and "left" are not absolute; they are relative. In the same way, "up" and "down" are also relative and not absolute. Up to a person in the north pole is different from what it means to a person in the equator or in the south pole.
This page was last updated on July 18, 2015.