What bible translation did the Unitarians use in 19th century America?

Emerson used the KJV. It looks like Channing probably also used the KJV.

Ralph Waldo Emerson

All Scriptural references in this study, unless it is specifically stated otherwise, are based on the King James Version, not because it is considered the best version, but because it was the one used by Emerson.

Source: Emerson's Use of the Bible by Harriet Rodgers Zink, January 1935 (page 4)

William Ellery Channing

I will base my for Channing upon the idea that the Bible translation that an author quotes is sufficient proof of "using" it. Therefore I will dissect tiny differences in translations. However, it is possible that these discrepancies could be unintentionally introduced by the author, e.g. from quoting by memory. Thank you to Nigel J for the Channing sources.

Letter on Catholicism: To the Editor of the "Western Messenger," Louisville, Kentucky (source):

Has not Jesus said, "When thou prayest, go into thy closet, and shut thy door, and pray to thy Father, who seeth in secret?"

This quotation is from Matthew 6:6, and when I search it in Google in quotation marks, the only thing that comes up is the letter from Channing himself! Perhaps, again, he was quoting from memory.

Belsham's Unitarian New Testament:

But thou, when thou prayest, enter into thy closet, and when thou hast shut thy door, pray to thy Father that is in secret: and thy Father, that seeth in secret, shall reward thee [openly].

King James Bible (source includes many other translations):

But thou, when thou prayest, enter into thy closet, and when thou hast shut thy door, pray to thy Father which is in secret; and thy Father which seeth in secret shall reward thee openly.

Translation: Inconclusive

Christianity: A Rational Religion

For example, the famous passage on which the Trinity mainly rests, "There are three that bear record in heaven, the Father, the Word, and the Holy Ghost, and these three are one," — this text, I say, though found at present in John's Epistle, and read in our churches, has been pronounced by the ablest critics a forgery; and a vast majority of the educated ministers of this country are satisfied that it is not a part of Scripture. Suffer no man, then, to select texts for you as decisive of religious controversies. Read the whole record for yourselves, and possess yourselves of its general import.

(Given the quotation and being a Unitarian, I wanted to give a bit of context!)

This quotation is from 1 John 5:7. I'll include verse 8 because the Belsham goes a bit off the goop on this one.

King James Bible (source includes many other translations):

For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one. And there are three that bear witness in earth, the Spirit, and the water, and the blood: and these three agree in one.

In particular, that "bear record in heaven" distinguishes the KJV (though perhaps not completely uniquely).

Belsham's Unitarian New Testament:

And it is the spirit which beareth testimony; for the spirit is truth*. For there are three who bear testimony, the spirit, and the water, and the blood: and these three agree in one.

* The received text reads, "For there are three that bear record in heaven, the Father, the Word, and the Holy Spirit, and these three are one. And there are three that bear witness in earth."

The footnote in the Belsham then has 9 notes about why they believe that text to be illegitimate.

Translation: KJV for use of "Holy Ghost"? But not a great example given the Unitarian slant.

Preaching Christ: Discourse at the Ordination of the Rev. John Emory Abbot, Salem, 1815 (excerpt):

"Christ, the hope of glory to the Gentiles," was the theme on which Paul, the Apostle of the Gentiles, delighted to expatiate. Having spoken of Jesus in this character, he immediately adds, "Whom we preach, warning every man, and teaching every man in all wisdom, that we may present every man perfect in Christ Jesus." [...]

The first quotation is actually a paraphrase of Colossians 1:27, but the second quotation is from Colossians 1:28.

Belsham's Unitarian New Testament:

whom we preach, admonishing every man, and teaching [every man,] with all wisdom; that we may present every man perfect in Christ

King James Bible (source includes many other translations):

Whom we preach, warning every man, and teaching every man in all wisdom; that we may present every man perfect in Christ Jesus:

Translation: KJV