A corner where two pieces of drywall come together will typically have a vinyl (shown below) or metal corner bead installed to to provide some protection.

The vertical crack probably just needs some caulk.

You could apply joint compound, sand and paint for a simple fix that may or may not last -- depending on how hard and how often that corner gets bumped.

That horizontal surface of drywall will always be a problem. Get a small piece of countertop material made that fits that with a 1" overhang. A shop selling Corian (lighter than granite, but you could use granite or any of the other top materials) could make one from cutoff. Glue it down with silicone caulk, then repair the drywall with a little mud.  

I would love some more context around this piece. Is it the only one in the home? How does it relate to the rest of the room? Is it meant to be a plant shelf under the window? Are there shelves inside the box?

You could use some flexible, paintable caulk on the long crack from bottom to top. I had drywall return windows in my previous home

It was a 27 year old tract home and most of the lower sills had cracks and damage. My painter replaced the lower sills with simple wood sills. something like this, but painted to match the white walls:

It was one of the best improvements I made in the 20 years I lived in that home. It was a change that many of my neighbors copied.

Drywall is meant for vertical surfaces or ceilings, not for table tops or flat surfaces that are going to be used to set anything on. I would consider replacing the entire top with something more suitable. Wood, granite, quartz, whatever fits the overall design of the home.

The others are correct; this is an inappropriate use of drywall. Have a solid surface fabricator make you a top with a drop edge that will cover most of the damage and prevent future damage.