Edit: the additional requirement of 'child safety' invalidates this answer. Fabric fastened at one end does not make the stairs safer.
If your requirement is cosmetic only: Carpet or fabric can be used on its own, but you'd have to fasten it on one side only: attach it to the bottom of each step, and leave the other end free.
When you walk up the stairs, sometimes you'll bump into the carpet/fabric. When you attach the fabric on both ends, your foot'll push out the fabric and cause it to tear around its fasteners.
If you want a solution that's rigid, you'll have to place a wooden board in each gap, thick enough to withstand the occasional kick.
For a removable solution, I'd use cloth. The type of fabric used for curtains would be a good option. Apply a strip of velcro to one edge of the cloth, glue the other half of the velcro underneath each step.
In the professional audio/video/stage production world, cloth skirts are used for a similar purpose: these are applied e.g. to the edge of a temporary stage to hide the underside of the stage from view.
To child-proof the stairs, the standard solution is to install a baby gate at the top and bottom of the stairs. By the time the toddler can walk up/down the stairs, he/she will be big enough not to fit between the steps.