I think that the service adviser probably meant one of the two weep holes on the front right side of the engine when they said "breather hose". Most likely the "round hole". Here's a link to a post that shows the two weep holes and gives an explanation of what they are for and what the significance of coolant coming out of them is:
ST1300 - Water Pump Weep Holes.
The ST motorcycles don't have a "breather hose" component on them. The later production ST 1300s (including your model year) do have a
drain hose attached to the "round hole". It's not a breather... the drain hose just routes any leakage down to the ground beneath your bike, rather than letting the leakage run down the side of the engine.
The only other possible source that the service adviser might have been referring to would be the overflow hose on the coolant reservoir tank (located on the left side of the motorcycle, directly ahead of your ankle when you are sitting on the bike), but I think that is unlikely.
Michael