I received a nice little rosemary Christmas tree as a gift. I was cooking chicken and decided to add some when I noticed it was covered with some white powdery dust. It seems that my little tree had powdery mildew.
Rosemary grown indoors is very prone to this and the little Christmas trees especially so. I investigated the cure and found that potassium bicarbonate products and neem oils can be used to control this disease. However, I was cautioned to make sure the product of choice can be used both indoors and on edible crops. Read the directions on the label and follow them perfectly since it is being used for cooking.
There has been some success reported with baking soda and water: spray with a solution of 1 tablespoon baking soda dissolved in 1 quart of water. Repeat if necessary. Here's a link to the Missouri Botanic Gardens helpful information.
I personally just tossed the plant in the dust bin and put it out of its misery!