The quoted language comes from Mencken's "The Professions," first published in The Smart Set (January 1922).
According to The Random House Historical Dictionary of American Slang (1994), one meaning of fan as a slang term is "to beat or strike." That meaning goes back more than two centuries and was certainly in active use in the early 1900s:
fan v. 1.a. to beat or strike, esp. repeatedly or soundly; deliver a blow to, esp. with a club.
[Examples:] 1785 Grose Vulgar Tongue: To Fan. To beat any one. I fanned him sweetly; I beat him heartily. 1839–40 Cobb Green Hand I 61: Let the loblolly boy be in readiness with the bite of a rope, to fan him to sleep. 1905 in "O. Henry" Works 1671: Out of th' park, now, for yours, or I'll fan yez. 1913 in J. Reed Young Man 31: "Chop it!" rumbled the cop, waving his club suggestively at me, "Now git along, or I'll fan ye!" 1917 Fornaro Purgatory 43: A guard "fans"him over the back with a club. 1928 Sharpe Chicago May 65: Go along with you, or I'll fan ye with my club. [Later citations omitted.]
It is interesting—and probably not coincidental—that all four examples from the period 1905–1928 involve policemen or guards threatening to beat or actually beating someone and using fan as a euphemism. The metaphorical "pantaloon fanning" in Mencken's article attributes a similar punitive role to university trustees, but this time the relationship evoked seems to be that of an adult threatening to inflict corporal punishment on a child. Harold Wentworth & Stuart Flexner, Dictionary of American Slang (1960) confirm this specific meaning of the term fan:
fan ... v.t. ... 2 To spank. 1931 : "And don't let him out of your sight, or I'll fan your tail!" Queen, Dutch Shoe, 65.
Since pantaloons is simply an old-fashioned way to say "pants" or "trousers," the image here is of the professor being swatted on the backside with a paddle or an open hand, as if he were a naughty or otherwise disobedient child.