。。
Leave it to me. You can rest easy, like you're safely aboard a great ship.
Literal
I [to-に] entrusted. You [topic-は] big-ship [on-に] rode [as-if-つもり] [with-で] being, is-good.
A boastful, reassuring offer. 任せたんだ — the past 任せる ('I entrusted / you entrusted to me') in the explanatory んだ form is a set way of saying 'it's in my hands now.' 大船に乗ったつもり ('as if you're on a big ship') is a fixed idiom meaning 'be completely at ease, have no worries.' つもり here is 'as-if' rather than 'intention' — the 'in the mindset of.' ~といい is the 'it's good if X / you should X' suggestion pattern. Confident, charismatic speech.