where do you live, and what zone are you?

I'm in Utah, I'm not sure what zone. I'm in the Salt Lake City area.

Put them under shop lights or t5s in your garage... I keep mine in a grow tent all winter in my basement and they do great! You can harvest all year.

herbs do not go dormant.

Also why are we talking about winter right now? Seriously depressing lol

Hmm... Some herbs do not go dormant. But some do.

I don't really have an answer, Alex. I'm assuming that you want to store them away somewhere, rather than just leave them on the balcony...?

I don't get frost where I live, and my mint thrives in winter, so I've never dealt with the dormancy stage of mint. But those garage temperatures sound pretty warm... I reckon even if you get it to die back to the rhizome, it could start up again while it's in the dark. Because it's just a rhizome at that point, I would assume that dormancy is dependent on temperature rather than day length. But it will survive being left out on the balcony, if that's all you're worried about-- you won't have to buy a new plant next year.

It might work with the sage. You're best waiting for somebody else's advice though.

I have several kinds of mint in 20" plastic pots on my front porch. The plastic pots are nestled in larger decorative concrete urns. The mint dies back each fall and then emerges each spring. So if you don't mind it going dormant, I'd say give it a try. If your balcony gets a lot of wind exposure, you may want to consider moving the pots into a more sheltered corner or cover with a blanket or tarp. In the early spring clip as much of the dead stems back as you can, remove any sprouting weeds, and give it a good watering. Also mint gets kind of territorial, so plan on giving it a pot all to itself.

I also have a regular sage plant in a 20" plastic pot. In the fall, after the first frost, I prune it by removing about 1/3 of the woody stems, mostly in an attempt to keep it from getting so tall that the pot blows over in the wind. After pruning, I move the pot into my garage, which to be fair is a good deal colder than yours, as in it doesn't freeze but during the coldest part of the winter it can get down into the upper 30s. I water maybe 1X a month max. Then, in spring when I notice leaves emerging, I give it a good deep water and a dose of generic liquid fertilizer and put it out on the porch.

I have several types of other herbs in containers in my backyard that stay outside all year long. They tend to die back in the winter and then re-emerge in the spring. You might try that. Things that come back: thyme, oregano, chives. Parsley and cilantro are excellent at reseeding themselves so there's another option. One caveat is to really make sure you have good quality potting soil.

My main concern with the garage is that the temps are high enough that plants will want to grow, and unless you plan on providing them light they will get weak and leggy very quickly.

What winter temperatures do you expect? Sage is a pretty hardy evergreen shrub. It is evolved to rest in the winter but not to die back or go dormant.

Mint, on the other hand, is a herbaceous perennial evolved in its natural habitat to die down to the ground in winter. You can keep it growing all year round if it has a lot of light and some warmth but it is prey to aphids on the soft leaves and stems.