You can have a go at teaching "threshold boundaries". This means training your doggo to automatically wait at doors and gates before going through (only when you give permission). If you search Google and YouTube for terms like "teach dog not to run out of door", you'll find lots of articles/videos on how to do it.
It takes a lot of time and patience. Realistically can take a year / until he's a more mature adult.
I'm working on it with my dog right now, as we have a gate blocking off the hallway to the bedrooms (as yes, he loves to go in and steal socks out of the laundry bin haha). He's actually started to hesitate when we walk through the gate without closing it. It can be done, just not overnight.
Basically - you start by teaching a "sit" and "wait" or "stay" command. If your dog already knows this, you simply use that command as you open the gate. You may have to start by just opening the gate a touch. Then a bit more. Until you can walk through the gate and back again, without him moving.
If he moves, you just return him to his sit and wait position and try again.
It won't happen in a day. So just aim for little moments of success.
Use really high value rewards for this, which you don't use anywhere else. Cooked chicken, mature cheese, hot dogs, that kind of thing. Or a ball/toy, if he loves that.
Meanwhile, do what you can to make the upstairs as unappealing as possible.
Another thing you could consider is a double gate system, which will remind your kids.
So, you'd have a gate at the bottom of the stairs, then another gate at the top of the stairs. Once the kids hit the second gate, it should remind them to go back and close the first gate - or at least prevent the dog from getting all the way up the stairs.
I hope that helps!