I would ask the landlord if you could install a gate that is anchored. I wouldn’t put a tension gate above stairs so the landlord would need to be ok with putting holes in the stairs post and/ or wall.
Our home has stairs like this. We have this baby gate that attaches to the railing and the wall. On the railing side, there's a piece of wood secured to it that can eventually be taken off to protect the railing itself. I think we got that separate. https://a.co/d/dN7IrmY
We lived in a 2 story house for years with all 3 of our kiddos. You just have to find the gate that works best for you. Practice walking up and down them with your kiddo. For example when they were learning to walk we showed them how to crawl up and go down on their butts. As they got older and more coordinated they learned to walk up and down holding the railing. There were also rules, no throwing things up and down the stairs, no trying to slide down them, no running, etc