Given the easterly direction of their travel across the Arabian peninsula, their departure from the shore had to also be in an easterly direction. We may therefore be quite confident that Lehi's party left the Old World heading east.
Nibley cites the Hilton's study of the Arabian journey of the Lehites, and discusses the departure from the peninsula:
"He says here, "In Salalah we confirmed the fact that the monsoons, which fill the Qara Mountains with life-giving moisture during the summer, also provide Salalah with a trade wind that could have taken the ship toward the Pacific" (p. 114), the trade winds which the Arabs discovered and used in ancient times in the sixth century. They go from the Northeast in the fall and winter, and then they come from the Southwest. This is the one they would follow in the spring and summer. And when they discovered the trades, they could go one way. They were prevailing winds; they kept going. All during the season they would take you this way from Malibar, from India, etc., and from the other half of the world they take you back again. So they could import the treasures of the Orient, and this is what Columbus was after, among other things." (Hugh Nibley, Teachings of the Book of Mormon, Semester 1, p.232)
"The Hiltons tell us here the "Arab entrepreneurs were sailing their dhows all the way from the Arabian peninsula to China. Arab ships rode the monsoons to the Malibar Coast of India, then on to Ceylon in time to catch the summer monsoon (June to September) and speed across the often treacherous Bay of Bengal, past the Nicobar Islands, through the Malacca Straits, and into the South China Sea. From here they were able to make a quick, if risky, thirty-day run up the main trading station at Canton in China. The trip from the Arabian peninsula to China took approximately 120 days." Now once they emerged from the Malacca Straits into the open, they could go the southern route or they could go the northern route. The Jaredites went the northern route, and they [Lehi's family] probably went the southern. Sometimes blown completely off course, they "would end up in the Pacific 'where, the Chinese believed, the drain spout of the world's ocean sucked the unwary sailor into oblivion'" (pp. 114-15) (Hugh Nibley, Teachings of the Book of Mormon, Semester 1, p.232).