One of the more common areas of a leak is the under the carb bank hoses and the aluminum joint where they attach to the head. The aluminum elbows can rot and crack as well as have an O ring seal failure. Also, the hoses are exposed to a lot of heat that can cause failure. It could be a relatively minor leak but you don’t want to have a more major failure when on a longer ride away from home. There are some very good write ups here on how to do the work and I’ve completed the work on each of my 2 STs in the past few months (never did it before and found it all pretty intuitive). It’s actually pretty straightforward if you take your time and are willing to do a little disassembly - if you choose to do the work yourself, I would get all the parts together before doing anything (new hoses, elbows, O rings, etc.). Also, just for completeness, it can be the overflow tube as it tends to get a little dry and brittle vs the bike needing a new radiator cap. Does it run hot? A thermostat could be faulty if that’s the case.
If you need it, I can look up my Partzilla list of the pieces I bought when I overhauled my STs
Edit...here’s the list:
Order Items
Ordered Shipped Total Price
16211-MT3-000 - INSULATOR, CARB
8 8 $70.72
19503-MT3-000 - HOSE, RADIATOR
2 2 $24.06
19504-MT3-000 - HOSE, L. THERMOSTAT
2 2 $35.64
19505-MT3-000 - HOSE B, WATER
2 2 $23.68
19506-MT3-000 - HOSE, R. THERMOSTAT
2 2 $30.00
19523-MT3-000 - JOINT, WATER
4 4 $33.32
91331-PC9-003 - O-RING (21.2X2.4)
4
Remember, the above is for 2 bikes. Also, I got some 1/4” fuel line for the overflow hose from a local Advance Auto parts, so no need to get this from Partzilla.