No just when engine running.
If an Alpha drive, The raw water hose from the transom to the engine cooling system may be leaking at the transom fitting or the hose itself, since the raw water pump is in the drive lower unit. But the Y pipe would leak all the time, since it is below the water line.
If a Bravo, and it's not leaking while sitting in the water and engine off, the leak is NOT below the water line (ie: NOT the Y pipe at the transom connection), since the raw water pump is attached to the engine.
It has to be on something above the water line (or an Alpha drive) for that drip to be present only while running