During winter time, if you are growing your curry leaf plant inside of the house, you need to check the plant every day preferably for the spider mites.
The spider mites breed so fast in the dry and warm environment, especially where no natural enemies are around. They make spider web like nest around the growing points and between the leaves also.
You can take the pot and shower the plant to wash them away, or use your fingers to take the webs and mash the spider mites if you do not mind doing so. There is Nontoxic starch-based contact pesticide available in some countries. You might be interested in looking for. In Japan, there is one called Nen-Chaku-Kun 粘着くん available at gardening nursery store.