Sick Curry Leaf (Murraya koenigii) Plants

I have two curry leaf plants (Murraya koenigii) which were given to me as a gift. One is a small leaf variety and the other is a large, dark green leaf plant. Neither is doing well in my house. In the winter time the plants are developing one type of diesease or an other. This last winter one plant had sticky droplets and we also found these sticky drippings on the carpet also. The other plant routinely has a very fine white web what looks like a spider web. What is a safe way to treat these plants to get rid of these diseases?

The fine white web is a tell-tale sign of a severe spider mite infestation. Curry leaf is highly susceptible to spider mites and constant vigilance is a necessary first step in keeping your plants free of these pests.
Bringing a heavy infestation of spider mites under control requires patience and diligence. Forgetting to keep up a cycle of treatment can cause the infestation to rebound and you have to start from the beginning again.
Chemical controls such as Pentac are effective on spider mites, but they are not safe to use on edible plants. Insecticidal soap products such as the Safer's brand that is widely available in North America will work also, but you need to repeat treatments for at least four weeks, and possibly longer – as long as 8-12 weeks.
The best way to apply insecticidal soap is to prepare a bath solution in a sink or pail using soap concentrate in water mixed according to the same directions for making spray. Then submerge the entire abound ground portion of the plants in the mixture, swishing the foliage around to ensure that all surfaces, and thus all adult mites, are exposed to the soap mixture. Because the mites congregate on the undersides of the leaves, spraying is largely ineffective.
Apply the soap treatment at least once a week for four weeks. It pays to invest in a hand magnifier of, say, 10X power so you can see the mites. Check especially the undersides of the leaves. As long as you see mites keep up the weekly treatment program. If you see a lot of mite activity a day or two after treatment, you can increase the frequency of treatment to twice weekly.
Because a heavy infestation means that there are lots of eggs, it is necessary to keep up the treatments at least two weeks after you last saw mites.

I would like to replace them or get one more plant but would first like to learn how to grow and take care of them. Can you help please?

Curry Leaf plants are hard to find. The seeds are very short-lived and they rarely survive the trip from India to North America. And, for U.S. curry leaf enthusiasts, the USDA prohibits the importation of seed or plants into the United States.

Also, I didn't find these plants in your online catalog even though I saw an email in the Q&A section which mentions someone buying this plant from you.

We were fortunate enough to have a one-time source of fresh seeds and from those we produced a few thousand plant which we sold in 2000 and 2001 but we are now sold out. We continue to look for a source of fresh seeds but without luck so far.