Walter was born about 1785 to Archibald Fairbairn and Alison Crosser, and died in 1859.
abt May 1785
Northam, Northumberland, England 
Birth estimates included:
abt 1784 Swinton, Berwickshire, Scotland (1851 census and headstone),
abt 1787 (death cert. says 72 in 1859)
1786 to 1791, England (1841 census)
so with the discovery of the baptism entry (thank you to a FamilySearch contributor) I think we can say about May 1785, Norham. Henderson-2297 23:20, 28 December 2018 (UTC)
22 May 1785
Norham, Northumberland, England 
where a sibling can also be found recorded two years later
19 Jan 1859
Morebattle, Roxburghshire, Scotland
cert. reads: Walter Fairbairn, pauper formerly ag lab, married, died 10:30am at Morebattle, aged 72 (h/stone says 75), s/o Archibald Fairbairn ag lab, dec, and Alison m.s. Crosser, dec; of natural causes, body seen by Henry ?Best? surgeon Kelso as certified by ??, bur. Morebattle Churchyard; Inf. Mary Fairbairn, dtr, present 
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It may be possible to confirm family relationships with Walter by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
Y-chromosome DNA test-takers in his direct paternal line on WikiTree: