According to the Wikipedia article on the song, Percy Montross wrote the lyrics "Oh My Darling, Clementine" in 1884. It seems extremely likely that Montross picked up the phrase "peak and pine" either directly or indirectly from Shakespeare, who uses it in The Tragedie of Macbeth (1606/1623):
1 [First Witch]. I my selfe haue all the other [winds], / And the very Ports they blow, / All the Quarters that they know, / I' th' Ship-mans Card. / Ile dreyne him [a sailor, whose wife had refused to give the witch some of her chestnuts] drie as Hay: / Sleepe shall neyther Night nor Day / Hang vpon his Pent-house Lid: / He shall liue a man forbid: / Wearie Seu'nights, nine times nine, / Shall he dwindle, peake, and pine: / Though his Barke cannot be lost, / Yet it shall be Tempest-tost.
Alexander Schmidt, Shakespeare Lexicon and Quotation Dictionary (1875) offers this entry for peak:
Peak, vb. ... to grow lean, to fall away: shall he dwindle, p. and pine, Mcb. I, 3, 23.
In the same lexicon, Schmidt identifies two meanings of pine as an intransitive verb in Shakespeare's works:
Pine, vb. 1) intr. a) to want food, to starve: [citations omitted] ... With for, = to hunger for: [citations omitted] b) to wear away, to languish: [citations omitted] With for, to languish for: [citations omitted].
Schmidt assigns definition 1(b) to pine as used in the "dwindle, peak, and pine" example from Macbeth, meaning that he views the phrase "peak and pine" there as meaning "grow lean or fall away and wear away or languish." Montross seems to have essentially the same meanings in mind in his lines "When the miner forty-niner, / Soon began to peak and pine" from "Oh My Darling, Clementine."
James Murray, A New English Dictionary on Historical Principles (1905) has this note related to "peak and pine" under a linger entry for peak as a verb:
peak v. {Found early in the 16th c., origin uncertain.} ... 4. ? To droop in health and spirits, waste away ; 'to look sickly' (J[ohnson]) or emaciated. Chiefly in peak and pine, a Shakespearean expression repeated by many later writers, chiefly as emphasizing pine. [Citations omitted.]
So it appears that numerous authors in the nineteenth century and earlier treated "peak and pine" as a set phrase conveying the idea of languishing or wasting away. In the United States today, however, at least in my experience, "peak and pine" is not in widespread idiomatic use.