The role of the GG in the birth of the Kathakali genre in Kerala is a discussed matter. Quoting Rich Freeman (“Genre and Society: The Literary  Culture of Premodern Kerala”, in Pollock ed. 2003, p. 487):

‘While the form of th[e] original Sanskrit work itself has suggested that it was a performed piece, it was certainly put to this function in Kerala, where the work was known as the Aṣṭapadi. It was apparently adopted in this form by the kings of Calicut for performance in the Guruvāyūr and other Kṛṣṇa temples, where it developed from songs sung by temple servants before the stairs leading to the sanctum (hence, stair-song, sōpāna-gīti) into a performance genre called aṣṭapadiyāṭṭam (dancing of the Aṣṭapadi) presented by Cākyārs using the gestural language (mudra) adapted from the theater. The best guess is that this genre was developed in the fifteenth or sixteenth centuries, when the Caitanya movement in Bengal (which had direct ties to south India) may have given an impetus to the Kṛṣṇa cult in Kerala.’ [or a new impetus, since there was already in Kerala a strong Kṛṣṇa cult and lyrico-devotional tradition, with Bilvamangala etc., from which Caitanya himself borrowed a lot].

At the least the GG has clearly influenced the Kṛṣṇagīti, a lyrico-dramatic poem written by Mānaveda (a Zamorin of Calicut) in 1652 (cf. for the dating chronogram, Kunjunni Raja, CSKL p. 102) which became the source-text for the Kṛṣṇāṭṭam, a special dance-drama (different from the Rāmaṇāṭṭam viewed as the more direct predecessor of the Kathakali in the same 17th century).

It is to be noted that a verse form the GG is always sung at the beginning of a Kathakali performance as an auspicious ritual (see Paul Martin-Dubost, Le Théâtre dansé du Kérala, Paris, 1990, pp. 103-123).

The earliest translation of the GG in Malayalam seems the Bhāṣāṣṭapadi, by the poet Rāmapurattu Vāryar (1703-1753).

From A. Gaur, Catalogue of Malayalam Books in the British Museum, 1971, col. 110, these two references:

• dēvagīta… gītagōvindattiṉṯe svatantraparibhāṣa. Dēva-gīta, a free translation in Malayalam verse by Caṅṅampuḻa Kṛṣṇa Piḷḷa, with introduction by the translator, Trichur : Mangalodayam Press, 1946.

• maṇipravāḷāṣṭapadi athavā laḷitagītagōvindaṃ. Maṇipravāḷāṣṭapadi, or Laḷita-gīta-gōvindaṃ, a rendering in maṇipravāḷam verse, by K. V. Rāma Vāriyar, Ottapalam : Kamalalaya Press, 1921.

The second one could correspond to Rāmapurattu Vāryar’s work.

There must exist several other translations in Malayalam.

Below, a few references from a quick check with a local on-line bookseller (note that the second item is an edition of Rāmapurattu Vāryar’s work).

Testifying for the immense popularity of the GG in Kerala : in the Malayam novel The Legend of Khasak (khasākkiṉṯe itihāsaṃ) by O. V. Vijayan (1969, https://en.wikipedia.org/wiki/Khasakkinte_Itihasam ), at the end of the chapter 9 examined by Francis Zimmermann (http://ginger.tessitures.org/malayalam/fiction-contemporaine/khasak/ ) a anonymous Sanskrit stanza is quoted, which is, as I myself noticed, nothing else than GG 1.16.