Beyond the fire/flame, the biggest hazard with a gas forge is.... they produce huge amounts of carbon monoxide. Some type of fresh air is a must. Unless fully outdoors, I'd recommend a carbon monoxide detector/alarm.
The hidden danger with a gas forge, is the gas/propane, and LEAKS! I have been witness to everything from seeing all the windows blown out of a shop, to a building bursting into flames from gas/propane leaks. Don't ever take for granted that there are no leaks.....until YOU prove it with a spray bottle of soapy water. I don't check every time, but have a reminder on my phone for a once a week check for leaks. For, that means from the 500 gal tank located about 30' behind the hot shop....all the way to the forges it feeds. For me, leaks always seem to occur when there is a drastic temp swing from hot to cold. Which happens fairly often in Montana. Not unusual to see it be in the 50s.... and then a few hours later, below zero. So.....schedule yourself checks for leaks..... and always check after replacing a tank, or doing anything to any fuel lines and/or fuel fittings on a forge.
When it comes to the fire hazard, most who've not deal with a gas forge, don't realize the perimeter needed... generally 6+ feet away from anything flammable, and further from combustibles, and even then you have to be aware of where hot scale or bits of hot steel fly away to. The part that people don't get, is the duration of the heat. What I mean is that you fire up a forge, and it usually runs for HOURS, and in a situation that might not be dangerous with a short duration flame or heat...... you could have something several feet away start smoldering/scorching without noticing. Even in my shop, with a forge hood, and concrete floors, the walls surrounding the forge table/forges, is covered in 18ga sheet metal, from floor to about 9', and I am very keen on keeping an eye on things when I have the rear of any forge unblocked/open.
Basement? NO!!!
In a garage? Provided you have taken the effort to ensure things are safe....AND you have a big fan pushing fresh air in/out.
Outside? Likely the safest location, but it presents issues with seeing colors of heated steel, often resulting in overheating and/or burning things up.
Be Smart= Staying Safe.