In the Google view the position of the ferry site is obvious. On the north bank the track past "Ferry Cottage" heads for the river (and then bends away to the east). On the South bank a field track heads directly for Ferry Cottage (and then gives up).

Site of Gatehampton (Basildon) Ferry

Just below Gatehampton Railway Bridge at Runsford Hole
Basildon house is seen in the distance, and the church, rather a solitary one, is a short way from the tow-path below the railway.
891:  The name Runsford Hole appears.

John de Shelford is Lord of Gatehampton and Runford;  he and his ancestors from time immemorial had duas Lockas in the water of the Thames.

1777:  Oost Mill (Ouse, Oes)
1792: Picturesque Views on the Thames by Samuel Ireland -

THE village of Gathanton is a pleasing object on the Oxfordshire side the river, from whence we enter a beautiful enclosure, formed by the surroundlng woods.
The happy serenity of the approaching evening gives an air of solemnity to the scene, while the moon, faintly rising, gleams through the beautifully variegated trees. The profound stillness of the air was not unpleasantly disturbed by the largest flight of ravens, daws,&c. that I remember to have seen ; it consisted of some thousands, who nightly cross the river to take up their abode among the neighbouring woods.
AT a small distance from the river, on the Berkshire side, Sir Francis Sykes has erected a spacious mansion, which is secluded from the view by the intercepting hills. The house, though superbly furnished, is still wanting in the more elegant decorations of the fine arts, which have not yet gained admission.

1810:  Ferryman appointed.
1881: George Leslie, "Our River" -

The tow-path changes sides some way after the railway does, and there is the usual horse-boat and little ferry cabin.  The old man, Levi Collins, who formerly kept this ferry, is now at the one at Mill End, near Hambleden.
Just before the railway-bridge is reached there are some tremendously deep and sudden holes, no doubt made by the ballast dredging at the time the bridge was built.

1885: Dickens's Dictionary of the Thames:

Basildon, Berkshire, on the LEFT bank, a small village, nearly midway between Streatley and Pangbourne, and standing a little distance back from the river. Population about 700. On the hills above, and somewhat to the south-west is Basildon Park ...

1891:  Ferryhouse rebuilt
1892:  Basildon Ferry, Henry Taunt -

Basildon Ferry, Henry Taunt, 1892
Basildon Ferry, Henry Taunt, 1892
© Oxfordshire County Council Photographic Archive; HT6194

1920: Fred Thacker in Thames Highway says:

Gatehampton (Basildon) Ferry: A statement occurs in 3 Hen.IV that "John de Shelford is lord of Gatehampton and Runford; and he and his ancestors from time immemorial had duas Lockas in the water of Thames: one at Gatehampton and the other at Runford." The former is probably at Hart's Lock. see below; the latter is perhaps identical with Runsford: a meadow name at the junction of Streatley and Moulsford parishes on the River. The O.S. marks a Runsford Hole at the spot. The name is as old at least as A.D. 891. There were once two mills here upon the Basildon bank. Andrews' map of 1777 shews Oost mill, and another, near the present railway bridge; Oost being the more westerly. The latter was still standing in 1788; and possibly survived even as late as May 1840, when the towpath "near Ouse Mill" was being repaired. A variant of the spelling is Oes. An Ouse Mill House appears in Bartholomew's modern Thames map. This mill is variously stated to have stood near The Grotto at Basildon. Basildon towpath meadow names of Gasson, Great and Lock meads occur about 1842. The Commissioners' improvements had arrived at this point by the summer of 1787; and a ferry was contemplated. I get, however, no further history until the ordering of a ferry boat in August, and the appointment of Chas. Emmett as ferryman in September, 1810. Jas. Hall occurs in charge in October 1822, with 24s. wages; doubtless monthly. Thomas Costiff was ferryman in 1854; in addition to his work at Cleeve and Goring locks. No further name occurs until John Sheppard under the Conservancy in October 1866, with £2 monthly: possibly son of Wm. Sheppard, the Mapledurham lockkeeper in 1828. John retired injured in February 1875; in September Williams his successor (possibly identical with the Goring lockkeeper noted in 1869) was discharged in favour of Henry Belcher. His "wooden hut" was in a very bad state; and was to be rebuilt at a cost of £25. He was found drowned in February 1879; and G. Piggott had his place. Leslie says that Levi Collins, at Aston ferry in 1879, had previously been ferryman here; but the official records afford no evidence of this statement. In November of this year Mr. Sworder the landowner obtained compensation "for foot passengers put over by the Conservancy men"; and the Authority consequently ordered that the use of the ferry was to be confined to persons engaged in the navigation. Several petitions ensued for the restoration of public user; but the petitioners were recommended to secure first the consent of both riparian landowners. A similar agitation, with similar result, occurred in 1884. Immediately afterwards the millionaire landowner of Basildon sought permission for himself and his household to use the ferry, to the continued exclusion of the general public; and was very properly refused. Rebuilding of the ferryhouse was being arranged in May 1891; northwest of the old site. Cooke has a view of apparently the original building, published in 1811. C. Bossom had charge of the ferry after Piggott, who died in 1880. The former was pensioned in August 1894; and was succeeded by T. Bossom, his son. This son was dead just before March 1897; and the father died, perhaps from shock, in the following month. T. Collier followed them at the ferry.

[ I don't know when the ferry ceased.]