Have you tipped it out of it's pot to look at the roots? I'm hoping for lots of white ones, but if all you see are brown ones, the roots may have died. if you see no roots circling round the soil's perimeter, gently comb away (with a wooden skewer or toothpick, for example) to see if there are roots in that soil.

Gently feel the stem part you say is browning. Is it firm, or squishy?

I could very well be wrong, but is this a scented leaf geranium? The plant looks like a fresh cutting to me that maybe hasn't developed a sufficient root system yet.

Keep out of direct sunlight while it is in this droopy state, and please report back on the condition of it's roots. Great that you included a photo too!