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The phrase damsel in distress designates the clichéd literary topos of a young woman (usually of high social status) who is in trouble and needs a man’s help.

The earliest occurrences of the phrase damsel in distress that I have found are as follows, in chronological order:

1-: From An Account of the Greatest English Poets. To Mr. H. S. April 3, 1694., by the English essayist, poet, playwright and politician Joseph Addison (1672-1719), published in The Fourth Part of Miscellany Poems. Containing Variety of New Translations of the Ancient Poets: Together with Several Original Poems (London: Printed for Jacob Tonson, 1716) [page 288]—the reference is to the English poet Edmund Spenser (c.1552-1599), best known for The Faerie Queene:

Old Spencer next, warm’d with Poetick Rage,
In antick Tales amus’d a barb’rous Age;
An Age that yet uncultivate [sic] and rude,
Where-e’er the Poet’s Fancy led, pursu’d
Thro’ pathless Fields, and unfrequented Floods,
To Dens of Dragons, and enchanted Woods.
But now the mystick Tale, that pleas’d of Yore,
Can charm an understanding Age no more;
The long-spun Allegories fulsom grow,
While the dull Moral lyes too plain below.
We view well-pleas’d at distance all the Sights
Of Arms and Palfries, Battels, Fields and Fights,
And Damsels in Distress, and Courteous Knights.
But when we look too near, the Shades decay,
And all the pleasing Lan-skip fades away.

2-: From The Fair Nun. A Tale, by the English poet Elijah Fenton (1683-1730), published in Miscellaneous Poems and Translations. By Several Hands (London: Printed for Bernard Lintott, 1712) [page 216]:

But as she in her Cell lay sighing,
Distracted, weeping, drooping, dying,
The Fiend, (who never wants Address
To succour Damsels in Distress)
Appearing, told her he perceiv’d
The fatal Cause for which she griev’d;
But promis’d her en Cavalier,
She shou’d be freed from all her Fear.

3-: From The Adventures of Gil Blas of Santillane. A New Translation, by the Author of Roderick Random (London: Printed for J. Osborn, 1751), by the Scottish novelist Tobias Smollett (1721-1771) [Vol. 2, Book 5, chapter 2, page 197]:

Alphonso said, “Gentlemen, that lady and cavalier whom the robbers have bound to trees, are perhaps persons of the first quality: and shall we suffer them to fall victims to the barbarity and brutality of thieves? Take my advice, let us attack these banditti, and put them all to death.” “With all my heart, (said Don Raphael) I am as ready to do a good as a bad action.” Ambrose, on his part, signified his willingness to lend a hand to such a laudable enterprize, for which (said he) I foresee we shall be well recompensed. I dare likewise affirm, that on this occasion I was not at all afraid of the danger; and that no knight-errant ever shewed more readiness to succour damsels in distress.

4-: From The Adventures of Roderick Random (Dublin: Printed and Sold by Richard James, 1755), by the Scottish novelist Tobias Smollett (1721-1771) [Vol. 1, chapter 22, page 185]:

All night long my imagination formed a thousand ridiculous expectations: There was so much of knight-errantry in this gentleman’s coming to the relief of a damsel in distress, with whom he immediately became enamoured, that all I had read of love and chivalry recurred to my fancy, and I looked upon myself as a princess in some region of romance.

5-: From The History and Adventures of the Renowned Don Quixote (London: Printed for A. Millar, T. Osborn, T. and T. Longman, C. Hitch and L. Hawes, J. Hodges, and J. and J. Rivington, 1755), translated from the Spanish of Miguel de Cervantes (1547-1616) by the Scottish novelist Tobias Smollett (1721-1771) [Vol. 1, Book 3, chapter 12, page 180 & chapter 13, page 181]:

The curate fell upon a scheme, extremely well adapted to the taste of the knight, as well as to their purpose. He proposed to cloath himself in the dress of a lady-errant, and that the barber should disguise himself as well as he could, in the likeness of a squire; which being done, they should go to the place where Don Quixote was, and the priest, on pretence of being a damsel in distress, should beg a boon, which he, as a valiant knight-errant, could not help granting: this boon should be a request, that he would accompany her to a certain place whither she would conduct him, there to redress an injury she had received from a discourteous knight […].
[…]
[…] Scarce had they sallied from the inn, when the curate began to think he was to blame for disguising himself; it being in his opinion, indecent for a priest to appear in such a maner [sic], how much so-ever depended upon their success. He therefore proposed, that he should exchange characters with the barber, who might act the part of the damsel in distress, while he took that of the squire, which he thought, did not so much profane the dignity of the cloth.