Temperatures from 65 - 80°F (18 - 26°C) are ideal during spring and summer. Try around 55 - 60°F (13 - 15°C) for a month or two during winter (before spring) but no lower than 45°F (7°C) which may encourage blooms for late spring - summer.
Plenty of bright light is needed during the active growing season. Direct sunlight during the day is fine for a couple of hours or so, but do avoid hot summer sun. A spot close to a window with the right balance of sunlight and shade is ideal.
While the plant is actively growing you'll need to water the plant thoroughly (once the top soil has started to dry out). Just top the water up slightly during the winter, only when the first top inch of soil becomes quite dry. Rainwater or distilled is best used for watering, especially in hard water areas.
A peat moss based potting mix with 2 parts peat moss and 1 part perlite is suggested.
Repot once every 2 years.
During the summer extra humidity may need to be provided if the air becomes dry, with misting or a humidity tray. If you can, also avoid dry air from artificial heating during the winter.
Non flowering stem cuttings are best taken during summer, and need to have at least two nodes present to propagate. A propagator or pot with a plastic cover is required to provide enough humidity and heat (temperature needs to be about 70°F - 21°C or more). Bottom heat mats or heated propagators will improve your chance of success, and use rooting hormone on the cut.
The stephanotis can also be propagated by seed - if your lucky enough to have the pear shaped fruit appear (containing the seeds). Propagating seeds is a long process of waiting for the fruit to mature and preparing them for sowing, and then providing enough warmth for germination. Your most likely to be propagating using stem cuttings.
Not a lot of pruning should be done; it wont encourage new growth very much at all, unlike many other vines. Any shoots with leaves not doing so well can be removed during spring.