Ahhh that drain hole
Yep it's got nothing to do with your Alternator......when you get a coolant leak under your throttle bodies from 1 or both of the 2 short right angle hoses the go to each side of the engine it "appears" through here.....
This is the most common of the coolant weeps from the ST1300.....
Main reason it occurs is the honda hose clamps are very narrow and it's safe to say they are in a buggar of a spot and although Mr Honda workshop is supposed to check tightness at service time he in reality doesn't.....so a weep starts.
The solution is new hoses, new coolant while your going, new wider hose clamps all round and face the clamps in such a manner that you can check very occasional.....all though once done you probably won't need to ever again.
Loads of info here on the procedure
Cheers