The phrase reds under the bed and variants were used to denote an exaggerated or obsessive fear of the presence and harmful influence of communist sympathisers in a particular society, institution, etc.—cf. also better red than dead – better dead than red and meaning and origin of ‘booked any good Reds lately?’.

The earliest instance that I have found is from the Chicago Sunday Tribune (Chicago, Illinois) of 28^th September 1924:

DAVIS SCOFFS AT ‘MOSCOW PERIL’; HITS JOHN BULL
Calls Bob G.O.P.’s ‘Bogey Man.’
By Donald Ewing.

Wilmington, Del., Sept. 27.—[Special.]—John W. Davis, Democratic presidential nominee, tonight turned his attack on the British form of government with a declaration that it is “the British lion,” and not the bolshevist “Moscow wolf” that threatens American liberty.
The candidate was in the midst of a joint denunciation and defense of Robert M. La Follette when the startling statements against the nation to which he once was ambassador were made. He had attacked Charles G. Dawes* for spreading the belief that “under every bed is a Red.”
And then, turning to La Follette’s plan for congressional veto of the Supreme court and the constitution, he declared with a shout that the story that La Follette would lead us on to Moscow and bolshevism through this practice, was a Republican “campaign bogey man,” set up to delude voters.
                                                            “Back to London.”
“My real objection to Senator La Follette’s proposal is not that it is leading us on to Moscow, but that it is trying to take us back to London,” he said.
Vetoing the Supreme court, he declared, was just returning to the British form of government with a parliament supreme and a constitution that becomes at will what parliament wants it to be.

* Charles G. Dawes (1865-1951), Republican vice-presidential nominee in 1924 (read below)

This cartoon, published in the St. Louis Post-Dispatch (St. Louis, Missouri) on 7^th November 1924, represents a businessman, with a smile on his lips, getting into bed; the caption reads:

No reds under the bed now.

The cartoon alludes to the result of the 1924 presidential election: Calvin Coolidge (1872-1933), the Republican candidate, defeated both his Democratic opponent and the national leader of the progressive movement, Robert M. La Follette (1855-1925), already mentioned, who had split from the Republican Party to form the Progressive Party; this was the reason that, in the same issue, the St. Louis Post-Dispatch said:

Captains of industry and finance are gratified at the result of the election.