Water heater is leaking so we turned off the water to it. Should we shut off the power too until it's fixed?
The heater is leaking.
Therefore, it does not hold water.
Water heater electric heating elements die a short, nasty death if they are powered up in air, rather than in water. Gas water heaters will burn out the bottom if the flame comes on with no water in the heater.
You should shut off the electricity or gas BEFORE you shut off the water, or very, very shortly thereafter, both to reduce damage to the heater (which may not matter much if it's leaking in a way that means it needs to be entirely replaced) and also to reduce risk of fire when heat is applied to a water heater not full of water.
If you are in a place where your water heater might freeze you should STILL turn off the heat source (electric or gas), and then drain all the water from the tank (and consider what a bad idea it is to locate your water heater where it can freeze in the first place, but that's an opinion...)
For most repairs and any replacement you'll want to drain any remaining water (amount depending on the location of the leak) anyway, but that's easier if you wait until it's cooled down some.
To the water heater? Sure, go ahead, unless it's located somewhere the water can freeze and it's super cold. Sounds unlikely in October unless you live in Alaska and your water heater is in a garage or something.