Is a vertical crack in a wall stud okay as long as the wall isn't load bearing?
There is no need to replace it. Just put in couple of 2 inch screws to tighten it up. ( your could squirt some wood glue into the crack for good measure before hand ) 

We can not see further down the wall but in theory it is attached to the 2x4 of the wall next to it so it is not carrying any load by itself.