DS learned to walk on his 12 month birthday-weekend. He was crawling up stairs before that. At 6mo, he started to climb up (opposite of steep) stairs on his own, then later on he started climbing up our "normal" apartment stairs. Our apartment stairs are indoors, carpeted, but the type that are slatted steps, no backing(fronting?) to prevent kids from sliding through the steps and falling. he would crawl up those on his own, no issue.
the issue started when he decided he could go DOWN the stairs by himself. He's fallen exactly 2 times, down a small number of stairs(imagine my fright when he fell down the first set of plush carpeted stairs, I thought that when kids fall down stairs they ALWAYS get hurt, and I started freaking out.) These stairs were not our apartment stairs, and I think he was probably pushed/knocked by some other kid.
The stairs we have now are marble, and he has almost constant daytime access to them. one time he drove his bike down them(EEEK!) luckily, the stairs have a landing less than halfway down and he only fell down maybe 5 steps. I've had to catch him a few times from falling, but he is usually better at maintaining his balance if he is doing it on his own.
I don't usually worry if DS is on the marble stairs by himself. He has been walking for 10 months now(really, only that long? seems like forever) and I don't freak out about these things. When we walk together, I hold his hand, and make him hold the railing. Sometimes he will refuse to hold my hand, and I make him hold the railing instead. I wouldn't bat an eye at plush carpeted stairs(which he also goes up and down, at other people's house, usually without me in sight)
But I don't freak out about a lot of stuff. I'm also the parent who taught DS how to safely unplug and plug things into the outlet, so he wouldn't hurt himself, in lieu of buying childproofing that he couldn't penetrate. There will always be uncovered outlets somewhere, or stairs that you don't notice, and I choose to teach my DS how to do things on his own for when he decides to do it on his own anyways.