Cover the outside of your weather balloon with some kind of tarp or netting that will extend down below the balloon itself. This cover will attach to whatever camera or sensory equipment you intend to send up with your balloon.
Anchor your balloon with some kind of small but weighty object. This object could be a small disc or weight, for example. A series of one-pound or even three-pound barbells attached around the outer cover would suffice. Attach these objects to several points around the outside of the cover of your balloon. These should press against the balloon itself and keep it from spinning too much.
Attach the same kind of weights to whatever cords extend beneath your balloon or onto the box containing your camera or meteorological equipment. This will help stabilize your equipment so that instruments get accurate readings and the camera does not take blurry images.
Check your photos or readouts from your instruments after your balloon has landed. If the readings or photos are blurry, you will know that your balloon was not weighed down enough. Attach more weights, and try your flight again.