Your jasmine plant has scale
The first sign of scale is the clear sticky honeydew on the leaves/stems and surrounding area. The bumps on the leaves and stems can vary in size and color due to the stage of development. Underneath the bumps (which protect them from sprays) are eggs and crawlers. They eventually crawl out from the shell but are difficult to detect because they are clear in color. It is at this stage, that they are vulnerable to sprays, and if treated can be controlled.
Persistent spraying is important. Neem oil can be used twice a month for 2 months. Use 1 oz. (2Tbsp) of Neem Oil , 1 1/2 teaspoon of dishwashing detergent to 1 gallon of water. Another recommendation for shiny-leaved plants only is Cedoflora, which is low in toxicity, safe and effective.
A stronger option is Orthene, a systemic pesticide. It renders the whole plant as poisonous and insects die after feeding on it. It smells bad and should be used with extreme caution. (Never indoors). There are several systemic houseplant insect controls (B0nide, Bayer)available that can be used indoors for potted houseplants.
For all chemical products-make sure you follow manufacturers direction for safe and effective usage.
For a safe treatment try some horticultural oil on a Q-tip and dab on individual adult scale. The oil will kill them by suffocation. For cottony cushion scale use rubbing alcohol on a Q-tip. You can also scrape them off with your fingernail which can be a very tedious task. *Repeated sprays of insecticidal soap once a week can help control the problem. One treatment is not effective as you may kill the adult but not the next generation. Check your plant often for scale insects so that you can control any infestation before it gets out of control.
*Do not use insectidal soap on shiny leaved plants (gardenia); as they have a tendency to be sensitive to soap sprays which causes phytotoxicity-yellow/brown spotting and burned tips.