Kimchi “brine”

Per NC State's Extension's article on pickles and sauerkraut (some emphasis added):

Pickles or sauerkraut mold during fermentation.

Answer:

Unsafe—microorganisms are growing improperly.

Possible reasons

• Fermentation temperature was above 75°F.
• Too much salt was used, not allowing adequate lactic acid production.
• The cloth on top of the kraut was not kept clean during fermentation (may need to be replaced after skimming).

Per Penn State Extension's Saurkraut Guide (emphasis added):

Do not taste it if you see mold on the surface, feel a slimy texture, or smell a bad odor.

Their PDF on Saurkraut says:

To avoid surface mold growth, keep the cabbage submerged at all times by covering it with a plate just small enough to ft inside the fermentation container or with two or three clean quart jars filled with water. An acceptable alternative is to fill a large, sealed, food-grade plastic bag containing 4½ tablespoons of salt and 3 quarts of water.

Again, no indication of acceptable types of mold.

I cannot endorse the existing answers which indicate that mold is an acceptable part of the saurkraut making process, unless credible science based evidence is presented, from reputable sources such as University Extension Centers or, better yet, peer reviewed journal sources.

If your jars aren't in the refrigerator already, I highly recommend unscrewing the lids as soon as possible...unless you want to be able to share stories about how you found glass shards and the smell of kimchi everywhere in your kitchen one day.

Depending on when you mean to eat them, I'd recommend a mix of room temperature ripening and fridge storage. Keep the lid completely unscrewed while at room temp until ripened to a stage you prefer, and keep it loosely on in the fridge and eat it up in the next few weeks (where it will continue to ferment some more but slow down drastically). It's not a terrible idea to put something like a sheet pan under the jars just in case one overflows.

Cheesecloth is overkill, but you could choose to use that or anything that'll cover the top of a jar without making it airtight while at room temp. Glass jars are used often in commercial kimchi production and sale, so other than making sure the lid isn't fully screwed on they're fine to use. The explosion concern is mostly when you leave it too long (more than several hours?) at room temp or in the fridge for more than a couple weeks - while the jar is completely sealed.

This all depends on the kind of kimchi you are storing, of course. I'm assuming it's the freshly made napa cabbage type you're talking about. Some kimchis don't ferment as much or at all.