Anchor the chairs at the base of their legs using a sandbag or a couple of hefty rocks. Tie two bags of sand together with a piece of thick rope. If the chair has a cushion, carefully remove it. Lay this over the center of the chair seat, so that the sandbags are hanging from the side of the chair. Secure the rope in place with a couple of nails or industrial staples. Reattach the seat cushion. If there is no seat cushion, purchase a small pillow roughly the size and shape of the chair to cover the rope.
Using wood screws and strong fishing net, build small scaffolding on the underside of the chair seat. Attach the fishing net to the chair, securing the corners to the upper part of the leg. Fill the net with heavy stones that are too big to fall through the mesh. The chair will instantly gain weight, making it more stable.
Lay a heavy blanket over the top of the chair to prevent it from blowing over. This also has an added bonus of keeping your chair free of dust and debris. In addition to a heavy blanket large, you can lay heavy pillows on the seat top to weigh down the chair even more.
If you have a wooden deck and wooden chairs, fasten them together with a large wood screw planted through the base of the chair legs and into the deck. You might also try using an L-bracket or corner brace that will hold the legs of the chair more securely to the ground. (Ref. 2.) This will prevent the chair from being toppled over or stolen, although it will also prevent you from moving it around without a fair amount of work.
For a chair in a slightly wooded area, anchor the chair to the ground using an auger anchor. These are unique anchors that resemble large screws with only a few threads at the tip. Most of the length is smooth. The screws dig deep into the ground without disturbing the soil too much, anchor themselves as they're drilled into the dirt. Attach an L-bracket to the legs of the chair and drive the auger anchor through the holes on the ground. (Ref. 1.)