One of the best ways to help save animals from extinction is through education. There are many reasons why animals all over the world become extinct or are in threat of becoming extinct. Perhaps they are losing their habitat due to various reasons or are being poached but whatever the reason is it is important to get the word out that these animals may soon disappear forever. There are many clubs and organizations that are dedicated to saving animals from extinction. Your best bet would be to contact one of these organizations and find out what you can to to help.
While poaching (as mentioned in the first answer) is a major problem, the loss of habitat tends to be just as much, if not more, of a problem. Even so, it is unlikely that we in the West can really do anything about it.
The only way to prevent loss of habitat is to prevent deforestation. Since we do not live in Russia or China, we cannot do anything directly (by pressuring our legislatures) to stop deforestation. Russia and China are two countries that are not well known for giving in to outside pressure, either. So, in my mind, the forest habitat of the tigers needs to be protected but I am not confident that Westerners can do anything to protect it.
To some extent, globalization and globalized efforts might be the best approach on this subject. Recently, China and Russia have signed agreements to designate portions of their shared borders as a wildlife sanctuary for Siberian Tigers to reside. The belief is that such endeavors can allow for new tigers to grow and mature without fear of extinction, with a hope that the population numbers increase by early 2020. National legislation that forbids hunting the Siberian Tiger might be another step in this direction. It has already been taken, but ensuring that it is enforced would be another measure taken to stop any eradication of the species based on poaching.