While Stalin attempted to construct a tunnel under the Tartar strait with forced labour from the Gulags in eastern Siberia, construction was abandoned after a few kilometres had been completed, and while there is intent to finish the project eventually, no money is forthcoming and for now the only options are to sail or fly.
By plane
The booming oil industry has ensured an unusual abundance of options to reach a destination as remote, and sparsely populated, as Sakhalin. The airport in Yuzhno-Sakhalinsk has connections not only to major cities in the Russian Far East, but also flights to Japan, South Korea and China several times per week. If you are dubious about flying with a Russian airline, the only other option is the weekly Asiana flight from Seoul, South Korea.
By ship
There are several ferry routes connecting the mainland with Sakhalin, but unless you posses time, patience, and Russian skills in abundance, your choice is pretty much limited to the daily ferry service between Vanino on the mainland, and Kholmsk on the islands western coast. Vanino is linked with the rest of the Russian railway network by a daily service to Vladivostok, with stops in Komsomolsk and Khabarovsk en route. In the summer months another option is a Japanese operated ferry service linking Korsakov on the shore of Aniva Bay, at the southern tip of the island, with Wakkanai on the northern tip of Hokkaido.