What kind of sand should I use for a sandbox

Paint on a concrete walkway will never last very long, and adding sand would definitely make it chip faster. The correct way to do it would be to stain it. I would use a good stain instead of what you get at the big box stores. Then seal it. Concrete sealer you can get anywhere. You can add some sharkgrip to it for the non-slip you are looking for.

To clarify, this staircase is entirely brick, correct? If so, you absolutely don't want to use concrete. Concrete is harder than the brick and will eventually damage the bricks.

You want to use mortar. If it's an older staircase, it's likely sanded mortar (mortar + sand).