The yard appears to have been on the current location of Durham House Street and the eastern end of John Adam Street, according to this 18th century map.
The overlays that highlight 17th century London features are approximate and derived from Wenceslaus Hollar’s maps:
Open location in Google Maps: 51.509420, -0.122810
Durham Yard, Strand, on the river side and a part of the grounds of Durham House.
Durham Yard, anciently Duresme House, as being the residence of the Bishops of Durham. ... Of later times this Durham-yard came to Philip Earl of Pembroke and Montgomery, in consideration (say some) to pay to the see of Durham £200 per annum, which grant was confirmed by Act of Parliament, dated the l6th of Charles I. And it was by his son built into tenements or houses, as now they are standing, being a handsome street descending down out of the Strand.—Strype, B. vi. pp. 75. 76.
From some satirical verses, printed by Anthony a Wood, respecting Le Tellier, Archbishop and Duke of Rheims, who came to England in April 1677 to "treat about a marriage with the Lady Mary, daughter of the Duke of York, with the Dauphin," it would seem that even then Durham Yard was a place of questionable resort. For—
The bishop who from France came slowly o'er
Did go to Betty Beaulie's;
and this Betty, we are told in a note, was "an old bawd in Durham Yard." 3 In Dryden's Sir Martin Marr-all, the scene of which is laid in Covent Garden, Lady Dupe speaks of Durham Yard as if it were the usual landing-place for that neighbourhood; and in The Taller of June' 7, 1709, mention is made of "a certain lady who left her coach at the New Exchange door in the Strand, and whipt down Durham Yard into a boat with a young gentleman for Fox Hall." Sir Godfrey Kneller's first London residence was in Durham Yard.* David Garrick in his short-lived venture as a wine merchant had his "vaults" in this yard. His brother was his partner. "Foote used sarcastically to say that he remembered Garrick living in Durham Yard, with three quarts of vinegar in the cellar, calling himself a Wine Merchant"l During a part of the time that Garrick had his vault in Durham Yard his friend Johnson had his " garret in the Strand," at "the Black Boy over against Durham Yard."2 There was an earlier wine merchant than Garrick in Durham Yard, one Brinsden, whom Voltaire addresses as "dear John," wishes "good health and a quick sale of your Burgundy," and shows, by the general tenor of his letter, that in the bright springtime of his genius the great French writer must have been a frequent visitor at "durham's yard by charing cross."
Durham Yard is just below the NE corner of this map
Chart showing the number of references in each month of the diary’s entries.