Which prayer did sailors in Russian America expect to hear?

A tough but interesting nut to crack, I don't think I have the definitive answer, but I did find some promising leads.

Firstly, there's the Orthodox prayer of intercession:

O Lord, Lover of men,
forgive those who hate and wrong us.
Do good to those who do good.
Grant our brothers and relatives
their saving petitions and eternal life.
Visit the sick and frant them healing.
Guide those at sea. Travel with travellers. Struggle alongside the Orthodox.
To those who serve and are kind to us,
grant remission of sins. On those who have
charged us, unworthy as we are, to pray for them,
have mercy according to Thy great mercy.
Remember, O Lord, our fathers and brothers
who have fallen asleep, and grant them rest
where the light of Thy countenance shines.
Remember, O Lord, those who bear fruit
and do good works in Thy holy churches
and grant them their saving petitions
and eternal life. Remember also, O Lord,
us Thy humble, sinful and unworthy servants,
and enlighten our minds
with the light of knowledge of Thyself,
and guide us in the way of Thy commandments,
by the prayers of our immaculate Lady, Mother of
God, and Ever-Virgin Mary, and of all Thy Saints,
for Thou art blessed to the ages of the ages.
Amen.

However, I don't necessarily think that is the one you're looking for. In 'A Manual of Eastern Orthodox Prayers' this is linked to St. Ioanniki (d. 846), and used as a commemorative prayer to be said after evening prayers. The main strength for this option is the regularity with which this prayer would have been used.

This is because this US Navy article suggests that in the 18th century prayer at sea was only beginning to be formalized. It also links to the '1789 U.S. Book of Common Prayer' and 'Prayers to be Used at Sea.' which has numerous (Protestant) options. It is not impossible that some of these might have been inspired from the same psalms as the Russians would have used, but I'm afraid this is a fairly broad lead.

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Assuming a similar prayer would have been held on naval ships during the Russo-Japanese War, this might be helpful:

Everything was done in the solemn Russian style. On the eve of departure, prayers for a safe crossing had been held on the Suvorov and on the other ships as well.
_{—Shiba, 'Clouds Above the Hill', Vol. 2}