As I remember it the 0-200 (continental engine) has a carburettor mounted on a manifold above it with the Airbox below it. Some only had one priming line that went to the manifold just above the carburettor and not individually to manifold at the cylinders. If this is the case it is perfectly normal that you will see un-vaporised fuel dripping from the airbox, since gravity is at work here. It is particularly apparent when the weather is cold, since less of the fuel is able to vaporise. The fuel is dripping back down the carb and out the bottom. This could be a good thing since the excess fuel is escaping and should be less of a fire hazard than it might be if it did not. Once the engine starts the rest of the fuel will be quickly consumed by the engine.
These engines do not have a 'Choke' so the engine will be more difficult to start in cold conditions if you use ecessive throttle openings. Setting the throttle 1/4 open for example could be too much. Shut the throttle completely, prime the engine and start the motor with the throttle closed. The throttle butterfly will act as a valve and stop some of the fuel draining away, also it will act as a 'choke'. When the engine fires, that is the time to open the throttle to the 1/4 open position and it will quickly suck up the remaining fuel.