If engine is cold (hasn't been started) engine can be off, but if engine has been run or is running only add coolant or water with engine on. If it's leaking a lot water can be used until it is repaired. As soon as possible. NEVER add coolant or water to a warm or hot engine you Will cause major damage, unless engine is running.

It should be cool and off. Please have it repaired. if you overheat your engine, you can run up a serious repair bill ( $thousands) if you warp the head.

Also, you shouldn't use water. Water has a lower boiling point than coolant, and does not inhibit corrosion. If you are using coolant, you are leaking a fluid which might poison your little furry friends.

let the car set and cool down, it should be OFF

What??? Thermostats don't leak.

My car has a water leaks it's the thermostat leaking water when I put water in it should the car be running or not when I notice it I start running hot I stop and turn it off and out water in it should I leave it on?