Though Whitman wasn't directly associated with the movement, he is often lumped in with American Transcendentalists like Ralph Waldo Emerson and Henry David Thoreau. The Transcendentalists believed in a spiritual unity that underlies that natural world. We come to know about the Soul through nature. "Crossing Brooklyn Ferry" shares this view, and Whitman includes human commerce and industry as a part of nature. You can argue about whether his poetry is "religious" or not, but it's certainly spiritual. One of the reasons Whitman avoids specifically religious language is to be as inclusive as possible. He didn't like dividing people up into sects – like, oh, say, North and South, black and white. (Hint, hint: Whitman was writing around the time of the Civil War).
The speaker does not condone evil actions like lying, cheating, and adultery, but he accepts that they will always exist and thinks people should not feel so guilty about them.