The Oxford English Dictionary finds one use of “baby” to mean “A lover; (also) a spouse. As a term of endearment: darling, sweetheart” in 1684:

A. Behn Love-lett. between Noble-man & his Sister I. 328 Philander, who is not able to support the thought that any thing should afflict his lovely Baby, takes care from hour to hour to satisfie her tender doubting heart.

But they also note explicitly: After the isolated example in quot. 1684, the modern use of this sense originates in the United States.

The next usage is from 1862:

1862 H. M. Naglee Let. 8 Jan. in Love Life Brig. Gen. H. M. Naglee (1867) 119 Dear, dear, dear Baby, how often, how incessantly I think of you.

Then another shortly thereafter in 1867:

1869 H. B. Stowe Oldtown Folks xiv. 147 Yes, Tina, I am glad,..but, baby, we can't stop to say so much, because we must walk fast and get way..off before daylight.

Which indicates that it probably became widespread in the second half of the 19th century. Babe in general seems to go back to at least the 16th cenutry, but in this specific sense, it seems to date to the first part of the 20th century. OED’s first example is from 1911:

L. Hart Compl. Lyrics (1986) 4/1 Oh, ma babe, waltz with me, kid. Gee, you've got me off ma lid.

Interestingly babe meaning attractive woman seems to follow immediately after, with the first citation being in 1917, with the first citation meaning attractive man only coming in the 1970’s. Baby as a term of affection for a romantic partner follows a similar pattern, actually: the OED notes “In early use only applied to women”.

Language is interesting: babes, with the s, was used from the 1910’s as “In pl. Used as a familiar or affectionate form of address for a person of either sex.” Earliest example:

1918 Pacific Reporter 172 296/1 He returned to the apartment, ascended quietly, and whispered, ‘Hello, Babes!’

Last fun fact: the earliest OED citation of “babe-magnet” is from 1989:

1989 Washington Post 22 Feb. c1/4 The excitement is very much divided between the sexes... He's a babe magnet.

It seems to be built off of the earlier term “chick magnet” (which was first cited in 1970, but the second citation isn’t until 1989, meaning they may have hit the big time together).