Of British-English origin, the phrase to be left holding, or carrying, the baby, or (to be left) to hold, or to carry, the baby, means to have to deal with an unwelcome responsibility.
This phrase alludes to a stranger’s accidental (as opposed to a parent’s legal) responsibility for an infant—as illustrated by the following Reuter’s telegram, published in The Pall Mall Gazette (London) of Tuesday 23rd June 1874:
At Exeter a few days ago a woman, slipping a shilling into the hand of a boy, asked him to hold her baby for a minute and then hurried away. Finding she did not return the boy took his little charge to the police station, where it was found that the shilling was wrapped in a piece of paper containing an earnest appeal for the child to be temporarily taken care of.
The following from the Daily Mirror (London) of Monday 14th November 1960 reported the same type of event—with the title punning on the figurative and literal meanings of the phrase:
She is left holding the baby. .
A month-old baby was abandoned yesterday—in the arms of a woman cleaner at King’s Cross station.
A woman who had been feeding the child, a girl, in a waiting room, asked the cleaner to mind the baby while she “made a phone call.”
Then police got an anonymous call that a baby had been abandoned at the station.
They found the cleaner in the waiting room—holding the baby.
The child was taken to a council nursery at Kentish Town.
The earliest instances of the phrase that I have found indicate that the metaphor originated in money markets.
The earliest is from The Glasgow Herald (Glasgow, Lanarkshire) of Friday 26th February 1875, in an article explaining that one of the reasons for the rise in rents in Glasgow was “the mania which set in some time ago for speculating in house property”—the conclusion of the article is as follows:
There is an end to everything, however, and there is reason to believe that the progress of the property mania is drawing to a close. The shrewder of the operators are getting out quietly, but, in Stock Exchange phrase, some will be left to “hold the baby.”
The second-earliest instance of the phrase that I have found is from a letter by a person signing themself W. J., published in The Times (London) of Friday 20th December 1878; the purpose of the letter was to “utter a word of warning about the new and plausible Parliamentary scheme for the appointment of an arbitrator in the City of Glasgow Bank’s affairs”:
What would be the probable operation of this Bill? It could only be this—that under its powers the arbitrator could make and enforce calls more rapidly—the object being declared to be rapidity—than the liquidators would, and that creditors who are naturally in despair at the delay in getting their money, and to some of whom such delay may mean ruin, may squeeze it out at the earliest possible date and save themselves from bankruptcy. Let me suppose a case. Let me say that there are a number of bill discounters in London or elsewhere whose readiness to accept its paper without inquiry, in the confidence that some one else would be left ultimately to ‘hold the baby,’ was one main cause of the bank’s ruin, and that they are left holding it, to their great concern. The interests of such men have been all in the appointment of a creditors’ liquidator, such as some London houses applied for at Edinburgh, and failing that, in the promotion of such a Bill as that now given notice of.
Finally, The Western Daily Mercury (Plymouth, Devon) of Tuesday 12th March 1895 published the following from its own correspondent in Cape Town, South Africa:
Attention has been drawn off from politics by the most magnificent boom in gold mining shares that has ever taken place. All Johannesburg men tell me that there has never been anything so solid and genuine as the enormous profits made in the mining markets since last September. The present boom—for it is not yet over—differs entirely from that of 1888-9, which ended so disastrously for the men who were left to “carry the baby.”