To make the bottom and top of the balloon anchor, could I use fishing line?
To make this type of balloon anchor sculpture, you would have to create a frame first. For the middle pole you could probably use PVC conduit, as we show for the balloon column base.
For the bottom and top part of the anchor I would recommend to use aluminum rod, bent into shape for the anchor bottom.
I found a nice step-by-step video for a balloon anchor by Sandi Masori. The design is a bit different from the photo you uploaded, but the principle is the same. She makes the frame completely out of aluminium rod.