It looks like there is a litte moss or lichen growing on the older woody material. It will not affect the health of the vine, so as long as it is putting out healthy new leaves and flowers, then you don't need to do anything about it. Here is more information on lichens and mosses: https://extension.umd.edu/resource/lichen-algae-and-moss-trees/. I do notice that the leaves look a little yellow and have some leaf spots on them (possibly fungal). The leaves of star jasmine should be dark green in color if the vine is healthy. I recommend raking up and disposing of any fallen leaves and provide a little fertilizer to encourage new growth. You can use a fertilizer formulated for roses or flowers and apply it at half the label recommended rate in the early spring and again in late spring. Jasmine blooms on new growth, and it looks like you have a lot of older woody growth on the vine. You can lightly prune the vine to stimulate new growth, but wait until after flowering. Pruning will also open up the vine and improve air circulation so that leaves dry out quicker after a spring rain, which will reduce fungal issues as well as the lichen/moss. For more on maintenance and pruning of star jasmine, refer to: https://www.rhs.org.uk/plants/trachelospermum/growing-guide 

Thank you!