How can I make my balcony secure for my dog?

Can you post a photo? I would use a 4 x 8 sheet of lattice and use cable ties to fasten. Many use chicken wire, but I don't, as they can get a paw stuck and hurt themselves.

You can weave a fabric through the railings so he can't get through.

You could put a roll of chicken wire along the railings and tie it on with zip ties to prevent the dog from falling through. They also make this in a heavy plastic webbing sold as snow fencing etc. I get it at True Value Hardware..... comes in several colors... https://www.truevalue.com/catalogsearch/result/?q=plastic+fencing

I guess a lot depends on what the building will allow. If your dog is small and your concern is closing the openings in the railing, wire fencing in black will be the least visible option. If your dog is large and your concern is the dog going over the railing perhaps you could build planter boxes that are backed by a high lattice in a rigid frame. If you cannot install anchors onto the balcony above, you could become friends with the resident of the balcony above and secure some screening to the bottom of their railings. Love that you're looking out for you best bud.

My friend purchased black netting made for covering fruiting plants (to keep birds from eating the fruit) and fitted it to her balcony railing with zip ties. Another possibility would be to use plastic lattice, because it comes in a variety of colors and could be matched to your building's colors.

You could use strong netting like chicken wire or even plexiglass.

Sort of depends on how big the dog is. But big box stores sell a thick wire that can be cut to the desired size.