down if it ever gets below freezing. If left up and it freezes, the water's expansive force at around 33 degrees F is substantial enough to destroy stuff.
If left down and in the water, there is chance of corrosion. Severe corrosion depending on the water. Also chance of marine growth. Even in fresh water.
I've seen freshwater outboards that have spent their lives sitting at a boat house, in the water, rotted nearly completely away. Sometimes it has to do with electrical currents passing between the boat/motor and the dock that it is moored to. Sometimes it has to do with the Ph of the water itself. Sometimes it's a combination of the two. If your dock has electricity, make SURE it's done properly as I have seen things rot out in under 1 year because a dock was not wired properly. Improper wiring in the boat is also a concern.
If at all possible take it out of the water but I understand why sometimes that's not possible or not an option. Local lake, lot of guys have been installing/building lifts so that the boats are not sitting in the water at the dock/house. Some of them are crude; made out of a hand crank winch and some ropes or straps but they work.