It did used to be necessary to activate the booster before sending the ship. The boat also needed to complete the full journey and be back home before the booster expired, or its travel time reverted to the full time with no benefit whatever for the portion of the journey made with the booster. So, basically, we were screwed out of the benefit on both ends of the journey. I recall myself whining about that on the old forum.
That has changed. Last time I took the boat booster for a new task, I already had one ship out. When I activated the booster, it did shorten the time for all my boats.
I can't say what happens now if the booster expires while a ship is still on its way. Maybe next time I get one, I'll send a spare boat to Fructus near the end to see what happens.