I don't think just sanding it is enough. Maybe I need more than just sanding it, like stain or varnish?
Although the term has been diluted in recent years stain is for colouring wood, varnish for coating it for protection etc.
To finish this off and make it comfortable to use I would remove or carefully glue down any particularly large splinters, then give the whole piece a good sanding (probably best done with a random-orbit sander) followed by finishing it with a film finish, e.g. varnish or similar.
Some tips on sanding
Be careful when sanding not to round over the visible corners and edges too much*. I would recommend you sand the bottom edges more however, to give them a slight chamfer of at least 2mm or a little over a 1/16", which will help prevent splintering when the box is moved about. The same thing is commonly done on table and chair legs. If the box will be used on carpet you should make the chamfer wider (double if necessary) and be sure to varnish this bottom edge as well as the visible surfaces.
As your Comment above indicates you don't want the colour changed too much I think your best option in terms of finish is a waterbased polyurethane. I think you'll get find it easiest to get a good result if you use a roller to apply it (either foam or with a short knap). Three coats should be sufficient to give a smooth enough, cohesive surface that you can live with but don't be afraid to apply a couple of additional coats if the OSB looks like it needs it.
In addition to minimally changing the colour of wood waterbased coatings have some other advantages for you. The first is they are easily obtained, you don't need to go to a specialist supplier as any paint store or home centre should have at least a couple to pick from. Also they're fairly cheap as far as finishes go and of course cleanup couldn't be easier, you just use soap and water, which also saves money up front as you don't need to buy any solvents or brush cleaners to clean brushes or rollers. Last but not least they don't have a strong odour, something many people new to finishing find objectionable with solvent-based finishes is their pronounced smell especially as this can linger for hours or longer so you're not just dealing with it right at the time of application.
*You do want to soften them slightly (also referred to as 'relieving' or 'breaking' the edges) but it's easy to go too far and have something with nice clean lines like this end up looking a bit rounded, something generally frowned upon by woodworkers and taken as a sign of sloppy workmanship (unless it's clearly part of the look of a piece).
This relieving has two functions, the first is tactile in that a sharp edge in wood can be quite uncomfortable and a slight softening makes a big difference in how nice something is to handle. The second is to aid in finishing as finishes creep away from sharp edges when they're wet, leaving them only partially coated and prone to getting grubby over time.