Hi, You now know where in Ukraine the family came from
Have you searched for Naturalization papers?
They may show some personal information
Name: Date and Place of Birth. [on some show actual place name of the city/town/village.
Names of Parents including maiden name of the mother.
Port and Date of Departure - Port and Date of Arrival.
Name of Passenger ship.
Passenger list - may show other family members also aboard.
http://www.ukraine.com/ternopil-oblast/borshchiv/