Gaia Falls is the name of the location where Weyard's oceans pour into the void. The name Gaia Falls does not entirely represent the border between Weyard and the void, but is used to describe the massive waterfalls falling into the void. In Golden Sun: Dark Dawn, a mini-dungeon near the end of the game can be found carved into the Falls themselves: Gaia Falls Islet.
The oceans do not seem to run out of water despite the fact that Gaia Falls is a huge waterfall constantly pouring countless gallons of Weyard's water into the void. The currents near Gaia Falls are said to be very swift and very dangerous. Despite this fact, in Golden Sun: The Lost Age, your ship does not get pulled by a current nor can you sail your ship over the edge.
Absence of Alchemy
Because of Alchemy's disappearance, the Gaia Falls are constantly growing inwards as the world's edge crumbles and falls into the void. Many settlements are in imminent danger of falling into the void, including the Apojii Islands and the SW Atteka Islet, but more notably Prox (and critically, the Mars Lighthouse) in the north. The Proxians are aware of this and it is their motivation for their actions throughout the games. By lighting the Elemental Lighthouses, the power of Alchemy would be released and prevent Gaia Falls from growing, which is the primary goal of Felix and his companions in The Lost Age.
Return of Alchemy
Since the return of Alchemy, the continents have expanded and grown, as has the surface of Weyard as a whole: in an essence, Gaia Falls have retreated. Strangely, the world has also become layered, so that may continents and oceans sit below sea level, and Gaia Falls are not just endless waterfalls but also drop down to other layers. Floating islands have also appeared in the mists beyond the edges of the endless Gaia Falls, and a cave-complex in the falls has reappeared, where, in Weyard's ancient past, a piece of the Umbra Gear, the Umbra Cloak, was hidden. The Umbra Cloak was eventually discovered by Matthew and his group in order to access Apollo Sanctum.
The concept is similar to the belief held by ancient civilizations that the earth is flat. Until the discovery that the world was round, humans believed that, if they sailed far enough, they would simply drop off the world's edge.