The decline in the number of domestic servants does it mean that the market mechanism does not work—that the offer
of the going rate of wages does not secure a servant because servants
do not move to the highest bidder? This complaint—which would
be far from unanimous—would have economic significance because
it would imply that wages do not allocate servants: there is, however,
little empirical support for it.

From &quot;Domestic Servants in the United States, 1900-1940&quot; by George Stigler.
Did the number of domestic servants decline during the second half of the 20th century, and if so, what are the possible explanations for this decline?
Please post answers with academic references only.
I think that what Stigler called 'domestic servant' would be nowadays considered 'domestic worker'. For example, according to the Luebker, Oelz, &amp; Simonovsky (2013). Domestic workers across the world: Global and regional statistics and the extent of legal protection (a technically semi-academic source as it is a technical report but based on comments I believe this would be acceptable to you), implicitly equates domestic workers with servants (and provides statistics on domestic work). According to the report:

With respect to North America, national sources also show a relatively low share
of domestic workers in total employment – 0.5 per cent in the United States (2010)
... For the United States, the 2010 Current Population
Survey put the number of domestic workers at 667,000 men and women. This represented a substantial decrease, of roughly 140,000 workers, following the onset of the
economic crisis; the same survey counted more than 800,000 in the pre-crisis years
2005 to 2008. Job losses among domestic workers were therefore quite substantial –
an aspect that is not commonly highlighted in discussions of the employment impact
of the global economic and financial crisis. ...
Historical sources indicate that the number of domestic workers was substantially
higher at the time of industrialization in the United States. The censuses of 1870 and
1900 counted 975,000 and 1,455,000 “servants”, respectively (see Rubinow, 1906). [emphasis mine]

Note that the number reported above by Rubinow roughly match statistics reported by Stigler.
According to the report indeed the number of domestic worker ('servants') declined in the second half of 20th century as in 2005-2008 the number of these workers was roughly half of what it was in 1900.
The data sources for US are listed as:

U.S. Bureau of Labor Statistics, Current Population Survey 2010 (Labor Force Statistics: tables 17 and 18 at www.bls.gov).

Additionally they frequently reference data from published papers/reports/books like the Rubinow in the excerpt above.
As to the reason what is the explanation of this decline - the authors do not mention specifically reasons for the historic decline in the U.S. (although for U.K. they cite changes to immigration policy which in past allowed servants to get settled status after 5 years), but the authors and also sources they cite argue that historically most domestic work was done by; a) women, b)  immigrants.

There is a peculiar historical continuity with respect to the overrepresentation
of international migrants among domestic workers. Based on United States Census
data from the late nineteenth century, Rubinow observed (1906, p. 508) that “[a]s
most means of employment are closed to the foreign-born (even English- speaking)
women, almost every second woman is forced to become a servant”. At the time, the
largest numbers of migrant domestic workers in the United States came from Ireland
(195,000), Germany (161,000) and Sweden (45,800).

In the report they also argue that most of domestic work is still performed by migrants or people who face barriers into entry to other occupations. Hence, my first guess would be that the number of domestic servants declined because nowadays especially women but migrants as well face less restrictions when entering into other occupations (although to be clear this is my conjecture).