Here's a couple of pics of our staircase - not too disimilar to yours. We have the uprights as you can see but couldn't leave them open like your ones for safety reasons when the children were young. The uprights look very narrow in the picture but they're actually quite thick and made of red oak.
Your's is very similar. We considered putting something in the uprights on ours, but since our kids are grown and we have a pocket door at the top of the stairs, we decided to forget it. We originally planned to do a diagonal like yours using some type of cable running through holes in the 2x4s, but got lazy! If we ever sell the house, we might have to add something....although we're in a small township outside the city limits, not much rules around here. LOL Now, my brother's wife is expecting & they have the same thing. I don't know if I'd leave it with a little one in the house.