You're worried enough to keep a fire extinguisher on your deck, but not worried about using an outdoor propane burner inside your house?
I wouldn't, personally. Those types of propane burners are terribly inefficient, which means they tend to produce a lot of CO; I'm not convinced just having an open door and a fan going is going to give sufficient circulation. And if you don't... well, the good news is that you'll never know.
This is a matter that tends to generate a lot of debate, however. I just come down on the side of - it's a lot of risk to take (both CO and the fire from the burner) to keep yourself a little bit warmer.
What about setting the burner up outside, not too far from the basement door? Nothing to keep you from mashing in the basement. Keep boiling outside, but set up your burner so you can watch it from inside the doorway so you can stay out of the wind.
Besides, you're in Jersey, not Anchorage. I know it gets cold, but... c'mon. There are always those nice days even in February when it's sunny and not too windy, just put on your long underpants and an extra pair of socks and suck it up. I'd rather brew in 10° weather than 90°!