Spider mites
Specks barely larger than a pinhead, in tan, red, or pale yellow, clustered on the underside of rosemary's narrow needles. Fine webbing between needle whorls is the giveaway. A magnifier or a sheet of white paper held under a tapped stem reveals them.
Needles fade to a dull silvery-yellow then turn brown and drop from the bottom of the plant upward. Indoor rosemary in dry winter air can lose half its foliage in 2 to 3 weeks. The oil glands on the needles release a thin dusty bloom where mites have been feeding.
Rinse, then horticultural oil weekly for 3 weeks
Take the plant to the sink or shower and blast both sides of the foliage with cool water for 60 seconds.
Once dry, spray Bonide All Seasons Horticultural Oil (about ten dollars) on every surface, including the underside of every stem.
Repeat the oil spray every 5 to 7 days for 3 weeks to catch newly hatched mites.
Boost humidity and isolate from heat sources
Move the plant away from heat vents and sunny windows where air dries out. Group with other plants on a tray of pebbles and water, or run a small humidifier within a few feet. Spider mites thrive below 40 percent humidity and stall above 60 percent.