The OED says that the word is now local only, and suggests that one consult the English Dialect Dictionary.

It also says that second and third senses are current only in places where the Scandinavian influence was strong. That would seem to suggest that the first sense may still be current in certain other places than that, but still local.

staithe

[steɪð], sb. Now local.

Forms: 1 stæþ (dat. staþe, stæþe), 2 steþ, 3 staþe, 5-8 (9 Dicts.) stathe, 6 stath, 7 stayth, 8 steath(e, 6-9 staith, 6- staithe; also 9 dial. steeth, stay, etc.: see Eng. Dial. Dict.

Etymology: In sense ^(def#1), repr. OE. stæþ neut. (? once masc.) = OS. stad̶ bank, shore, OHG. stad masc., neut. (MHG. stat, inflected stad-, mod.G. dial. staa), Goth. staþa dat. (masc . or neut.):-OTeut. *staþo- f. *sta-: see stand v. In senses ^(def#2) and ^(def#3), which are not evidenced in OE. and are current only in districts where Scandinavian influence is strong, the word prob. represents (or has coalesced with) the cognate ONor. sto̧ð fem.(:-*staþwō) landing-stage (Norw. stød). Cf. also OHG. stado wk. masc. (MHG. stade, mod.G. dial. staden) and MHG. gistat neut. (mod.G. gestade) landing-place.

1. The land bordering on water, a bank, shore.

• C. 893 Ælfred Oros. i. i. §22 - Of ðæm mere ðe Truso standeð in staðe.
• O.E. Chron. (Parker MS.) an. 894, - Æt Butting tune on Sæferne staþe.
• A. 1000 Riddles iii. 6 (Gr.) - Streamas staþu beatað.
• A. 1000 Riddles xxiii. 19 - Brohte hwæþre beornas ofer burnan & hyra bloncan mid from stæðe heaum.
• C. 1050 Suppl. Ælfric’s Gloss. in Wr.-Wülcker 177 - Ripa, stæþ. [11.. Ibid. 546 steþ.]
• C. 1205 Lay. 7 - He wonede at Ernleȝe..vppen Seuarne staþe.

2. A landing-stage, wharf; esp. a waterside depôt for coals brought from the collieries for shipment, furnished with staging and shoots for loading vessels.

• 1338 Orig. Chartulary of Tinmouth Monastery 172 in Brand Hist. Newcastle (1789) II. 255 - Domus quam predictus prior et suus conventus..habent in predicta villa Novi Castri super le Stathes.
• C. 1390 in Gross Gild Merch. II. 169 [Lynn Regis] - Unam communem placeam vocatam le commen stathe cum pertinenciis.
• 1420 Eng. Misc. (Surtees) 17 - We, serchours of the masons and wryghtes of the cite of York..awarde and deme y^t a lyne be drawn straight fra ye corner of ye stathe of ye chauntery..un to ye nexte corner of ye stathe of ye common place.
• C. 1440 Promp . Parv. 473/1 - Stathe, waterys syde, stacio.
• 1519 in Archæologia XXV. 418 - For caryeng of y^e same ij cads [of Red Heryngs] to y^e Common Stathe, ij d.
• 1653 Lilburn Tryed & Cast 4 - [He] sold a thousand pounds worth of Coales that were upon the Staithes.
• 1667 Primatt City & C. Build. 26 - You may consider what Stayths or Wharffs there be upon the River.
• 1708 J. C. Compl. Collier (1845) 49 - The Rivers are not Navigable for Ships, so high as they Keys or Coal-Steaths.
• 1833 Ht. Martineau Tale of Tyne i. 1 - Train after train of coal-waggons slid by on the rail-road from the pit to the staithe.
• 1862 Smiles Engineers III. 11 - Arrived at the staiths, the waggons are emptied at once into the ships waiting alongside for cargo.
• 1905 Times 4 Mar. 9/6 - At midnight last night the River Tyne Commissioners’ new staithes..were totally destroyed by fire.

3. An embankment.

• 1698 De la Pryme Diary (Surtees) 185 - Their tyde..is fenced out with huge stathes, for if all the water might be suffered to come in that would, it would..dround..the whole Levels.
• 1839 Stonehouse Axholme 52 - The fertility of the soil..would induce th e inhabitants..by means of staiths and embankments, to reclaim the land thus formed.
• 1876 Whitby Gloss. s.v. Steeath, - Staithes, masonry to prevent the ground as a foundation from being washed away.

Staith was a word commonly used for the storage structures in railway goods yards for coals. It seems to have derived from staiths in maritime use. Brocketts glossary of north country words (1829) defines a staith as 'a place to lay up and to load coals at'. It seems to have been used as such from the early days of railways in the north east. It is commonly spelt with or without an 'e' on the end. Some websites list 'staith as having Anglo-Saxon roots whereas Staithe is listed as being of Danish origin. It no longer appears in many dictionaries and seems to be falling out of use, but there are many place names all the way down the east and onto the south coast incorporating the word in either form and so is still used as a place name, including pubs and restaurants.

In Yorkshire, on the rive Ouse, up to Goole, there are many 'Staithes'along the way. Marked on the charts, there is Crably Staith, Grove Staith, Bank House Staith, to name but a few. Interestingly the spellings are all without the 'e', (On ABP charts). As you pass the points, there is rarely a full landing stage or jetty, but there is evidence from a pile of rocks, that perhaps there used to be. It still seems like a North East of England description, influence by the Norse language.

The Stockton Coal Staithes were once noteworthy as being the destination of the Stockton and Darlington Railway, but I haven't heard the word used for anything else (apart from, as Andrew Leach mentioned, the village of Staithes). I didn't know until your question that this wasn't a North-East dialect word.

The Oxford Online Dictionary says

(in the north and east of England) a landing stage for loading or unloading cargo boats.