How do I fix wobbly half wall?

I had a similar problem.I used screws for the frame construction they are stonger and hold better. The part that s to the wall I made sure screwed to a stud. The plate I screwed to the floor( in my case i used expansion bolts it was concrete). I put large reinforcing angle brackets in all four corners. When I apllied the skin in my case plywood covered with formica it stopped any racking and was sturdy. Just an example it may not apply in your case but maybe any of these steps might help.