This definitely depends on the kid. My youngest can go up and down stairs with no problem, and he has been able to since 11 months. He learned how to go down stairs (backwards as otherwise noted here) very quickly - over christmas at Grandma's house, who has a single step up to the raised dining room. Nice short drop to practice on, he went up and down that repeatedly. We still have the gate up and accompany him up the stairs, but he's never fallen. He is part-crawling part-walking up the stairs (he's been walking for a month or so).
My oldest took longer to figure out going down, but he had it down by 15-16 months. We moved into a house with stairs at around this point. We gated the top, but not the bottom; he seemed to have a good handle of things. It also helped that we have a short landing with 2 stairs, then 9 stairs, then 3 stairs, at right angles, so there's never a long distance to fall.
Around 21 months, he had his only fall so far of more than just into the stairs, falling down about 4-5 stairs as a result of getting a bit too excited. He wasn't seriously hurt, just cried for a minute or two and then wanted to go back up. Stairs, fortunately, aren't all that dangerous in most circumstances to kids; they're not falling far enough due to their height to have major injuries. Not to say you don't need to be careful, but it's not the horrible demon a lot of people see it as; letting your kids have their head once in a while isn't likely to cause serious injury, as long as they know to be safe (and he definitely does, now). From our point of view, it was safer to let him practice on relatively safe stairs at home and risk a fall or two than to worry that he might run off while out somewhere and have a more serious fall on stairs that are unfamiliar and perhaps much longer, or concrete, or something else.
My advice is to let them practice on the stairs to their hearts' content. Make sure they practice going down backwards, but in general let them have controlled access to the stairs as much as you can until they are totally comfortable with them - and by controlled I don't mean handholding, I mean standing a few stairs down from them but otherwise giving them their head. This is the first of many things that you have to let them do to some extent even when there is some risk - in the next few months, if they're like my oldest, they'll be wanting to climb all over the park, and perhaps even the "big kids'" play equipment, which is clearly far superior to the little kids'. That will be much scarier than this, but it's all a part of growing up - learning to make mistakes while they don't cost you too much.
In terms of learning to descend, my youngest's method was pretty good for learning if you have it - a one or two stair descent that you can leave ungated. We left our bottom landing (2 stairs) ungated plus grandma's dining room, and between the two he learned on his own. If not, then the method we used with our oldest should work; take them to some stairs that you feel comfortable with, let them go up with you behind them, but instead of carrying them down, tell them they have to go down on their 'own'. Then, if they try to descent forwards, turn them around (so they reverse-crawl down the stairs). Keep turning them around until they figure it out. It might take a while - it took a few weeks with my oldest - but eventually they'll figure it out. The reward of getting to go back up the stairs is a pretty significant reward, after all.