When you say you are renting this house, do you mean:

(1) You own the house and are renting it out to other people

(2) You are renting this house from someone and want to make a temporary change to make it safe for your kids

Option 2

That’s actually a very good idea. Thanks for sharing

That railing is definitely not compliant with any current building code, and I agree it's very unsafe. I think the idea of zip-tying a barrier to it is good, but I'd use something more durable than window screening (and easier to attach). One product that comes to mind is plastic deer barrier fencing, like this: https://www.amazon.com/Barrier-Protects-Shrubs-Animals-Protected/dp/B000A16T7M/ref=sr_1_3?dchild=1&keywords=deer+fencing&qid=1597344519&sr=8-3

Even a gate at the top and bottom are going to be tough. It must be an older home. Current code says not more than 4” between balusters. I thought of zip tying snow fencing to it. Would look incredibly ugly.

Here are some ideas: https://www.pinterest.com/pin/239042692704558513/

Thank you for all the suggestions. Maybe i could make a plexiglass railing guard. But the Pinterest options are amazing as well.

Plexiglas would be nice, but are you prepared for the cost in a place you don’t own? Where I live, 1/8” thick plexiglas is about $5 a square foot, and you might want to even use 3/16” thick which would be about $7 a square foot.

If you can even find it right now as all of our suppliers are out of stock. Everyone has been using an awful lot of the acrylic for the Covid Screens.

Google bannister guards, they have mesh ones that are pretty cheap. My sister had to use them and they worked great.

This is what I used. Purchased from Amazon

Safety 1st Kids Safety Railnet for Indoor Balconies and Outdoor Decks, Extends up to 10', White

How old are the kids? Wouldn't be easier just to teach them not to pay on the stairs? IMO they would slip and fall faster on those narrow marble treads especially at the curve than hurting themselves on the railing. That curve most likely will not meet building code.