1) In the Old English period the word referred to a river bank or the shore: the OED notes that ‘staithe’ in the sense of a landing-stage is ‘current only in districts where Scandinavian influence is strong’.
The earliest Yorkshire references are minor place-names, starting with one in an undated thirteenth-century document: i.e. Quenstayth, Drax. The distinction in meaning is not always absolutely clear: 1283 culturam meam in … Overstathecroft, Acaster
1380 in the northern meadows of Snaythe on the east side del Stathe. On the other hand Hull had numerous staithes used by shipping from the fourteenth century: 1347 the common staith called Aldburgh stathe, Munkegate staith. In York and other inland ports the evidence is a little later: 1404 Resumptio stathe, pavimenti et inclusi, Selby
1434 pro reparacione stathe fracte super ripam Use, York. By the fourteenth century there were landing stages in Newcastle upon Tyne where coal might be loaded onto ships but the staithes which served collieries in Yorkshire were much later: 1778 the Town and Neighbourhood of Leeds have not, of late Years, been sufficiently supplied with Coals from the Coal-Staith
1833 Walker is going to drive a great trade and is … setting up coal staithes at Halifax, Bradford, and other places, Shibden. Such coal-staithes were erected later on sites away from the navigable rivers but accessible by wagon-ways, as at Staygate near Bradford, formerly Staithgate. The detailed accounts in this last case makes it clear that the staithe was built of stone.
2) As a verb the meaning ‘to strengthen a river wall’ was transferred to house-building.
1524 for two daysse taking downe the tymer off the for sayd howsse and staythyng the walls xijd, York.