Looking for something on winterizing an apartment balcony?

Hang a insulated drape ,however how tender are this plants and what is your location

I would go to Home Depot/Lowes and get a roll of the heaviest clear plastic I could find, and some 1x1" wooden firring strips, and then make that plastic as tight as could be on the strips into the edging around the balcony.

or you could also buy some sheets of the corrugated polycarbonate in the clear.... and use double-sided foam tape to secure them to the top, bottom and sides, and the panels as they join up.

Depending on where you live, you may have to bring them inside and place them in bedrooms, the bathroom, the kitchen, etc. Hawaii, Nevada, Florida and California, maybe not, New England, New York, Alaska, Montana, Colorado, almost definitely. Maybe before the frost comes you can move some to a friend's house or garage with grow lights?

Take measurements and consider all the other suggestions. If the deck floor has open cracks I would insulate the floor with cardboard with a rug on top. It looks like you already have a sheet covering your balcony. Switching it out with heavy mill plastic should be pretty simple. I would give it extra length and once secured at the top I would place it inside the deck railing to reduce access to winds and with the extra length weight it down. I would also insulate the inside of the deck railing by putting more cardboard right over the plastic, One possible way of securing the cardboard is with 3 or 4 tension curtain rods placed between the floor and the underside of the deck railing or any thing else that can be wedged in that space. I assume you're not allowed to use staples, tacks or nails.

Another option is setting up some sort of framework into which the plants could be clumped together and then covered by heavy plastic; a sort of a mini green house. Below is a creation that appears to be two tomato cages wired together.

The second photo, the tent, is 22" tall with a base of 22" X 22" and can be purchased from Amazon for $25. with free shipping. It appears there are different size options. https://www.amazon.com/Nuvue-Products-Transparent-Pop-Greenhouse/dp/B06XFNNL4H/ref=sr_1_sc_2?s=lawn-garden&ie=UTF8&qid=1509413727&sr=1-2-spell&keywords=nu+vue+pop-up+green+house

Good luck saving your plants. I can relate and I share your passion.

I'm in Austin, Texas. The winters are minimal (lowest 20's)

Heavy vinyl from Joann Fabrics, or other resource, and either a tension rod at the top, or a pipe if you can attach hooks for hanging. Then use the plants to weight down the bottom of the vinyl on the deck. There are glues you can use to make a pocket for the rod. or to attach it TO the rod. It can also be sewed, or stapled, if the rod is wood.