Okay, here's how to shut down Draven 101.
Without Q - Spinning Axe, Draven's damage is garbage. If you can properly play around this CD and mechanic of catching axes, Draven is much easier to deal with.
Because of the point above, Draven struggles heavily against reliable CC. Many supports can punish aggressive Dravens for catching axes in your face while trading, while others can prevent him from catching axes at all, simultaneously reducing his damage output greatly.
Heavy sustain lanes, and greater range, can outsustain Draven's mana pool. Champions like Caitlyn, Jinx, and Ashe can trade well from a distance while typically all running Warlord's Bloodlust, forcing Draven to epend more mana during trades for mobility. These champions also have decent zone control, making harder to do Draven things.
Remember, short trades (1-2 autos) will almost always be in Draven's favor, but extended trades/kiting will wear down any Draven.
Finally, Draven has to get a kill to cash in on stacks. If you force extended laning while not handing over first blood, you completely negate the entire reason behind the Draven pick.
He has a good win rate but an extremely low pick rate (on NA, at least). Caitlyn has a very, very high pick rate while maintaining a win rate within 1% of Draven.