There are no breathers to leak. It's a sealed, pressurized system, other than the overflow from the reserve (head) tank. But that shouldn't be puking coolant onto your bike's parts...it's designed to spit directly onto the street.
As Maverick said.....I think it's a 99 percent chance that it's one of the zillions of hose clamps under your fairings. A
famously loose one is at the crossover hose assembly, located under and to the left of the steering head area where three hoses come together, but there are
plenty of other locations that can slowly develop a leak after many hot/cold cycles and aging of the rubber. Snug them all. Some are quite hard to get to. You might need screwdriver bits or small hex sockets on a 1/4 inch ratchet....with assorted extensions and cuss words....to get them all.
Anybody know how many there really is? I think it might actually be less than a zillion, but not by much!
And don't go wild when tightening all those clamps. You might ruin the hose with too much tightening. Just snug them a bit and see if that stops the leak(s).