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Are you seeing any other symptoms? Normally, brown stems is an indication of stem rot fungus, which is common in soils that do not drain well. I would check your drainage and ensure that the plant is not in overly wet soil. If you catch this early enough, before the stems begin to turn blackish in color, you can improve the drainage (or move to a drier location) and prune out the infected stems/leaves. You can also treat with fungicide, though there is no guarantee this will cure the problem. Once the plant is too far gone, all you can do is pull it up, dispose of it and start again.

Height of 6 to 12 inches
Width of 6 to 12 inches

Plant in full sun location, average to sandy well draining soil.

Will not tolerate wet soil, allow to be on the dry side.

The culprit could be a nutrient imbalance: excessively high nitrogen in soil, or a deficiency of phosphorus or potassium. You may wish to get a soil test. If the plants are in a shady spot lack of light could be a contributing factor.

If you have an annual Dianthus, it will die over the winter so you can either remove them now or remove them next spring, whichever you prefer.

If you have a biennial or perennial Dianthus, do not cut back in the fall. The foliage from this year will help protect the plant's base from winter weather. The best time to prune these plants is just after they flower in the spring, as described in the Dianthus section in this article:

http://web.extension.illinois.edu/ccdms/yg/090831.html

This article has a list of annual and perennial Dianthus varieties:

https://www.gardeningknowhow.com/ornamental/flowers/dianthus/growing-dianthus-plants.htm

In north Florida, you would most likely have your first frost sometime in December. Most Dianthus flowers can withstand a light frost but not a hard frost. So they would most likely make it to December, but it's hard to predict whether they'd make it to January.

You can allow the plant to go dormant for the winter. I would store it in the same area as the geraniums.
Check to make sure the soil doesn't dry completely during the winter months. You can add a small amount of water each month.
I like to make sure the winter pots soil doesn't crack.
Adding some mulch on the top of the pot will help.

https://www.gardeningknowhow.com/ornamental/flowers/dianthus/growing-dianthus-plants.htm