In "Crossing Brooklyn Ferry," Whitman imagines that each subsequent traveler on the ferry would look into the water and see the same visions that he saw. “Closer yet I approach you... I consider’d long and seriously of you before you were born," he writes. In this and many other poems in Leaves of Grass, Whitman seems to be talking directly to you, the future reader of his poems. How does it feel to be directly addressed, especially by someone who lived in the nineteenth century?

This is a really interesting question. It's long been a tradition for poets to imagine their work still being read "many generations hence," as Whitman puts it in "Crossing Brooklyn Ferry." Think about Shakespeare's observations in his sonnets that the beauty of his beloved will not die so long as his work is read—"so long as men can breathe, or eyes can see." But Whitman is more direct in his approach, reaching out specifically to the imagined future reader with the "you" pronoun, which enables us to read the poem as if it had been written for us alone.
In terms of the theme of this particular poem, that's important, because Whitman is very much thinking here about continuity and the unifying experience of being human. Whitman imagines himself as being allied with others, and "it avails not, time nor place," because the unifying factor is our common experience. Whitman imagines others who might "look on the river and sky" and feel the same things he has felt. He is thinking, particularly, of the "others" many centuries in the future who will enjoy the same things he has enjoyed, the "sunset, the pouring-in of the flood-tide." Whitman's reference to the tides, in particular, underlines this sense that the human experience is actually as constant and as inevitable as the tide: just as nature will endure and the sunset we observe will be the same as those seen by people hundreds of years ago, the intrinsic experience of being human will always be the same, no matter the time or place.
Whitman's thesis in this poem is an embodiment of the idea that "no man is an island," as John Donne said. His preoccupation with the "ties between me and them," with "them" being the people who will come after him, forces us, as readers, to also contemplate the connections we have with the people around us in terms of time as well as space. Not only are we sharing experiences with the people who may be in a crowd alongside us whom we have never met, but we are also sharing experiences by being in the same physical place as others who are separated from us by time. That's a powerful concept, as is Whitman's statement that we, "you," are "more to" him than "you might suppose." Whitman, from the nineteenth century, reaches out directly to the modern reader in a way that makes us question how far we are really separated after all.