Minecraft wiki entry for Brewing, associated text: All ingredients for a single batch of any recipe can be loaded in order from a hopper for a hands-free brewing of a multi-step potion. (With the nether wart falling through to begin brewing.)
Since each step produces a "finished" potion (which can be taken by a bottom hopper, mass-production of potions is non-trivial. However, it can be managed if several brewing stands are chained together from bottom to side, each with fuel and stacks of the successive ingredients. The first step can be fed by a chest of water bottles.