I have a one and half year old who often opens the internal hinged door which would give him access to the stairs and outside. The door is nonlockable and we wouldn't want to it be lockable. Its second latch (?)- hidden metal thing part of doors mechanism that joins the door to side that keeps a door shut- is broken.
There are premade solutions for that situation, try searching the web for "toddler proof doorknob." Here's a random one from Target.
This one is designed for your style handle. Keep in mind that it will only work if the door handles turn independently. ie, can you open the door without moving the interior handle?
In a worst-case scenario you would need to replace the lever with a conventional round knob and place a childproof cage on one side. That's a simple project you can do on your own.
The best thing to do is fit a latch or lock up high. Anything to make the door take more force to open will make it more likely to shut on your small person and hurt them. It's also not likely to work very well.
As you need adults to be able to open the door from both sides, a bolt at the top isn't an option for you. A spline key bolt can be opened from both sides but needs a key. A normal door handle mounted unusually high is common in nurseries and other childcare settings and most doors (even hollow ones) have a suitable place. Alternatively you can sometimes find thumb turn both sides latches but these are rare. Fitting new a new door handle is a routine DIY job, but any carpenter or handyman could do it
Our solution was put a hook and tie a ribbon loop. Our child loves pulling the door ajar. He can pull ribbon off but his compulsiveness means he thinks of the ribbon as normal so he puts it back. We can take of the loop from either side of the door with some risk of him shutting door on our fingers.
