From Shorter Dictionary of Catch Phrases compiled by Rosalind Fergusson from the work of Eric Partridge & Paul Beale (London: Routledge, 1994), page 80:

keep your hand on your ha'penny   a piece of advice to an unmarried girl (from the use of the word ha'penny as a slang term for the female genitals); the full form is ...till the right man turns up. The phrase dates from around 1880 in this usage; during the 20th century it acquired the secondary meaning 'be careful or you'll find yourself expensively involved'. Both senses are now obsolescent. A variant of the phrase has threepenny bit in place of ha'penny.

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However, the oldest example I can find with Google Books is in The Darling Buds of May by H. E. Bates (London: Book Club, 1958), page 118:

'See you soon!' she waved.
'Any time,' Pop said. He laughed merrily. 'Don't do anything I wouldn't do. Keep your hand on your ha'penny.'

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So the proverb goes back to the 1880s, but I can't find any evidence that there was a song by that name before Alex Glasgow.