How to check if my 2007 Kia Rio base model cooling fan works?

Electric engine-cooling fan motors only run when the thermostat opens or coolant temperature reaches a set temperature.

So no, it doesn't run all the time.

edit: to check the operation of the cooling fan, with the radiator fixed, let the car run for 5-10 minutes, this should be sufficient for the engine to heat up and meet the criteria for the fan to start. Just keep an eye on the intrument-cluster for coolant temperature so that it doesn't overheat.