Remove support under the interior stairs
That stub wall does likely provide some support to stiffen the stairway. That said it would appear that the vertical stud that you circled in purple could be removed without any trouble. 

I would not suggest removing any other parts and leaving the horizontal double member in place. If you removed that it would leave a mess in regards to the already finished drywall joint that joint with the drywall in the bottom slope of the stairs. There appears to be minimal advantage for removing it anyway. In addition you will need that area to nail into for trimming the opening if you want to place an door to close off the space access.