Yes, it is considered healthy.
According to this study, even taking a few dozens of steps will improve your fitness, at least for men.
Furthermore, according to this study, several minutes of stair climbing will improve your health even more: "We know that sprint interval training works, but we were a bit surprised to see that the stair snacking approach was also effective," says Jonathan Little, assistant professor at UBC's Okanagan campus and study co-author. "Vigorously climbing a few flights of stairs on your coffee or bathroom break during the day seems to be enough to boost fitness in people who are otherwise sedentary."
And finally, as a result of the daily stair climbing you'll be able to climb more and more stairs. Therefore, you would perform better in a stair climbing test, which is the ultimate health test according to this article.
As far as for cartilage damage, if you aren't heavily overweight or suffering from arthritis or osteoporosis, you don't have to fear anything. People who get these kind of damages are often too heavy or overtraining a lot. In fact, climbing stairs or hills is actually even healthier than ground-level running for all your joints and cartilage, because the training effect per step is much higher and you are using more energy compared to running at ground-level. That's why experienced jogger like to climb hills, it's much less stress on your joints, combined with an increased training effect (although I'd suggest you to walk downwards, not running...yet here are some tips to run downhill better).
Note that it may look different if you'd be running up these 9 storeys, because in that case the risk of injuries might be increased, at least if you are not a well trained athlete.
As a final conclusion, I'd suggest to you walking up the stairs but not down in order to be on the safe side.