If you want to keep growing the Rosemary in water, you need to use a hydroponic system, which provides nutrition and deters infection, though Rosemary is more prone to botrytis when grown in this way; whilst Rosemary will root in water, it's usual for it then to be potted up into soil. Information on growing Rosemary hydroponically here https://gardeningtips.in/growing-rosemary-hydroponically-a-full-guide
UPDATE:
I suggest you plant it in new potting soil in a relatively small pot with drainage holes asap. Water in well,allow to drain down freely, and water ongoing when the surface of the soil feels dry (but not so dry it's shrunken from the sides of the pot),emptying out any outer tray or pot 30 minutes later. After you've potted it up, cut the stem back by a third to a half (see final sentence), to encourage new growth from the base. Hopefully,this will mean, although the browning leaves may fall (and when they do, remove them from the soil, don't leave them sitting on the top) your cutting will grow on healthily. The signs are hopeful, because there is evidence of new side shoots developing above the browning parts - cut back to just above a part with new side shoots.
As it gets bigger, you will need to move it into a larger pot, once it's formed a good root system in the soil.