There has been a lot of discussion about brewing in a basement. I have done some research on this and have successfully brewed 20 gallons of beer in my basement brew room in the last 60 days. I have not had any problems or issues, I am careful and pay close attention and never ever leave the the room with the burners lit. I do have CO Detectors and Fire Extinguishers and I do not store propane tanks in the use, when brewing is done the tank goes outside.
Here are the issues: Moisture, Fire, Carbon Monoxide and a build up of explosive gas.
With either with electric or propane you have to deal with the amount of moisture brewing produces.
With propane two more items, okay maybe three more. Those are: Carbon Monoxide, unburned propane and a propane tank in the house.
Moisture is easiest an exhaust fan like for a bathroom large enough to handle the cubic footage of the area your brewing in (may have to seek out a commercial grade fan) or a stove/range hood fan. Both will work well. The hood type will work better in a more open area like an unfinished basement. Either must exhaust to the exterior.
To use propane inside you must have burners rated for interior use, normally low pressure (of course they do not get as hot and take longer to boil the wort and heat strike water). Do not use the burners you have pictured in the house, in the basement or even next to the open basement door. Those are high pressure burners and not very efficient. They produce heat fast but throw a lot of CO and unburnt propane (an explosive gas when confined) into the air. Very dangerous inside.
If you have an enclosed room, like mine, an exhaust and an intake are needed. With an enclosed area heat can also be an issue as my room got a little warm in September, but over this last weekend it was nice as the air coming in was quite cold.
With your current set up, I would use the garage, and run hoses as long as the water flows it will not freeze (just drain them when your finished). Also if the temperature is above 25F I doubt the hoses would freeze unless they sat for many hours without running water in through them.
I would go for the garage, definitely safer!