The immediately obvious influences are Joy Division, Bauhaus and Siouxsie, all of whom are covered in the bands section.
However, amongst the punk and post-punk scene were several other bands who had definite influences on the early goth scene.
One such band were Adam and the Ants.
This may seem strange now, as Adam and the Ants are known mainly for their (or rather Adam's) later pop career, but before the pop career took off they were "art-school punk" with several parallels to the later goth scene, not least their emphasis on fetish imagery and use of sparse guitear and tribal drums. Certainly a fair few early goth bands seem to have been influenced by them, particularly Sex Gang Children and Southern Death Cult.
Mick Mercer kindly sent me some photos of the Ants which strikingly demonstrate their similarity in style to early goth bands.
Another band who may have had some influence are the little-known Gloria Mundi. According to Mick Mercer, they had a "goth" sound and imagery by about 1978. Their influence on other people and bands was probably quite limited as they were never very popular, but Mick maintains that Bauhaus underwent a serious image change after appearing on the same bill as Gloria Mundi.
A much better-known band who some people do consider as an influence were The Damned, but beyond Dave Vanian's image (and perhaps one of their later albums) I can see no real goth connection or influence at all (if anyone does know of definite connections or influences, email me)
(well, someone did email me, so thanks to James Lopez for this)
Further back, the pre-punk origins of goth can be argued endlessly, and have been. However, the most important influences on the early goth bands outside of punk were probably the Doors, the Velvet Underground and Bowie, as well as the 70s "glam" scene (T.Rex, Sweet, and perhaps Roxy Music).
Bowie in particular was very influential- apart from being a major influence on bands musically, he was responsible for much of the "glam" look and scene which influenced both goth and New Romantic.
And in 1974 he described "Diamond Dogs" as "gothic", something that may have had something to do with both Hannett and Siouxsie's use of the term.
Interestingly, since I wrote the above, I came across the following quote about Siouxsie from Nick Kwnt in the NME, 29/7/98:
Parallels and comparisons can now be drawn with gothic rock architects like The Doors and, certainly, early Velvet Underground.