The Kravats - Baby Let Me Take You Home (1965)

Blind Boy Fuller - Mama Let Me Lay It On You (1936)

(Traditional, sourced from Rev Gary Davis)
Influential version

Recording on ARC label by Fulton Allen.

Recorded a few months earlier by Walter Coleman whose version was issued later in the same year. 

See Bob Dylan’s Musical Roots (which questions the attribution of the song to Rev Gary Davis). Arnold Rypens’ The Originals discusses the song’s origins and lists some other versions and variations.

The Animals - Baby Let Me Take You Home (1964)

(Bert Russell - Wes Farrell, attrib. here to Price-Burdon-Valentine-Chandler-Steel [The Animals])
 UK
Influential version
#21 UK #57 USA

The Animals’ first single, on Columbia, March 1964, arranged by Animals keyboardist Alan Price, based on arrangements by Bert Berns. In the US the B-side I’m Gonna Take You Back To Walker became the A-side.

Chronology, and some speculation: The Mustangs’ version is often cited as the source of The Animals record (which sounds right from listening to the two tracks) but The Animals’ single was released a couple of months before The Mustangs’ single. Animals producer Micky Most brought Baby Let Me Take You Home back from a 1963 visit to New York where he consulted with Bert Berns. It is plausible that Berns acquainted him either with an unreleased Mustangs recording (released later in the wake of the Animals release), or with a demo recording, which possibly amounted to the same thing, considering that the Mustangs were Bert Berns and his collaborator Wes Farrell. 

The Bert Berns-produced singles by Hoagy Lands (below) and by The Mustangs have some key elements in common with the Animals version, notably the distinctive guitar introduction and an important melody change. The Mustangs single apparently uses the same backing track from Berns’s production of Hoagy Lands, but the feel is closer to The Animals, and thus closer to trends in beat music that were emerging from Britain at the time. Lands’s record, with vocals reminiscent of Sam Cooke, sounds more old-school, and his 30-second sung introduction is dropped on the Mustang’s record. The Animals drop it too, along with embellishments by a female vocal group, and they pare it down further by replacing a vocal line at the end of each verse with an instrumental break. The Mustangs and The Animals also share the fresh title, Baby Let Me Take You Home.

Certainly, in versions of the song from January-May 1964 by Hoagy Lands, The Animals and The Mustangs, the main melody has been tweaked by Berns enough to make it stand apart from the numerous earlier versions, right up to Bob Dylan’s (1962). In fact, a casual listener familiar only with, say, the Animals version might not recognise Baby Let Me Take You Home in a song such as Don’t Tear My Clothes from 1935.

Bob Dylan - Baby Let Me Follow You Down (1962)

(Trad., arr. Geno Foreman - Dave van Ronk - Eric von Schmidt - Bob Dylan)
 USA
Influential version

On his first album, Bob Dylan.

I first heard this from Ric von Schmidt. He lives in Cambridge. Ric is a blues guitarplayer. I met him one day on the green pastures of the Harvard University.

Dylan’s spoken introduction to the song

See commentary at Bob Dylan’s Musical Roots which casts doubts on the attribution of the song to Rev. Gary Davis.

Also recorded in the 60s by Rev. Gary Davis and by Dave van Ronk in 1964.

Hoagy Lands - Baby Let Me Hold Your Hand (1964)

(Bert Russell - Wes Farrell)
 USA
Influential version

Single on Atlantic, produced by Bert Berns (aka Bert Russell). The song edges towards the well-known version by The Animals, although it still had some changes to go through before then.

At around 0:30, after a slow introduction, , you can hear the distinctive guitar line that kicked off The Animals’ Baby Let Me Take You Home, released a couple of months later. The melody now takes a downward turn at the end of the first line, setting it apart from previous variations on the traditional song and making it recognisable to those familiar with The Animals’ version.

A current Beatles hit is echoed by the title Baby Let Me Hold Your Hand. Could I Wanna Hold Your Hand, taking off on the US charts when this single came out in January 1964, have reminded Berns of Professor Longhair’s 1957 record Baby Let Me Hold Your Hand (see below)?

The State Street Boys - Don’t Tear My Clothes (1935)

(Traditional, credited here to Willie Broonzie)
 USA
Influential version

78 rpm record on Vocalion, also released on Okeh, recorded with the participation of Big Bill Broonzy (guitar) and harmonicist Jazz Gillam. 

Kansas Joe & Memphis Minnie - Can I Do It For You? (1930)

(Traditional, credited here to M. McCoy - J. McCoy)
 USA
Influential version

78 rpm record on Vocalion, recorded in Memphis by husband-and-wife duo Wilbur “Kansas Joe” McCoy (1905-1950) and Lizzie McCoy, née Douglas (1997-1973), aka Memphis Minnie.

A distant ancestor of what became the Bob Dylan–Animals song. Memphis Minnie sings Do anything in this world I can… where Bob Dylan and The Animals sing I’ll do anything in this Godallmighty world…