This is a case where i would personally say the authenticity is somewhat dubious given the history of the period from which it is said to date and also the history of how and when it first came to light.

It might not surprise you to learn that the existence of this letter was first revealed by Khrushchev during his secret speech denouncing Stalin, which we know to have been riddled with falsifications and slander. For instance in the same speech Khrushchev referenced the so-called "Lenin's testament" which harshly criticized Stalin and called into question his suitability for leadership partly due to his "rudeness".

The problem is that this "testament" is almost certainly fake, as Lenin after his stroke was no longer in a condition to write himself by that time. It was Krupskaya who he supposedly dictated it to. Krupskaya however, though she was a great revolutionary and did a lot of good work for the Soviet Union in her official positions, is known to have had a bit of a personal grudge against Stalin at this point in time. (This is not a criticism of either of them, just a fact that sometimes personalities clash.)

While Stalin, for political reasons, never said anything about this, these documents were not signed by Lenin and there are numerous inconsistencies and problems with them (see links below).

So while this letter itself may be real and supposedly a copy of it is in the archives, it is not clear whether it was really Lenin himself who wrote it. At least he didn't and couldn't have signed it. It is also suspicious that the story told by anti-communist historians about how it was supposedly found on accident in Stalin's desk along with letters from Bukharin and Tito definitely does not stand up to scrutiny. There is absolutely no evidence of those other letters existing anywhere in the archives and they are clearly part of a narrative created to make Stalin look bad.

This letter about Krupskaya was officially published in 1989 by the then extremely anti-communist Soviet government under Gorbachev who was practically on a crusade to publish any and all slander against Stalin and anything that would portray the Soviet Union's past in a bad light. Some are genuine but many "admissions" and "uncovered documents" that appeared during that Glasnost & Perestroika period and afterwards under the bourgeois Russian government are very suspicious and probably fake.

Also, consider the circumstances that Lenin was in at that time. He would have found out about Stalin's rudeness toward Krupskaya through Krupskaya of course, who was one of his only contacts to the rest of the party when Lenin was gravely ill. Again i want to stress i have a lot of respect for her, and her revolutionary pedigree is not under any doubt, but she was also just human and of course when you have a personal beef with someone you are more prone to exaggerate their transgressions against you.

Also, here is a footnote from the Marxists archive about this letter giving some additional context:

After Lenin, with the permission of his doctors, had, on December 21, 1922, dictated a letter to Trotsky on the foreign trade monopoly (see this volume, Document 811), J. V. Stalin, whom a C.C. Plenum decision of December 18 had made personally responsible for the observance of the medical regimen ordered for Lenin, used offensive language against Nadezhda Krupskaya and threatened to take the case to the Control Commission for having taken down the said letter. On December 23, 1922, Krupskaya sent Kamenev a letter asking for protection from “the gross interference in my personal life, offensive language and threats”.

Nadezhda Krupskaya apparently told Lenin of this fact in early March 1923. Having learned about this Lenin dictated the document here published.

Maria Ulyanova later wrote in a letter to the presidium of the July (1926) Joint Plenum of the Central Committee and the Central Control Commission of the R.C.P.(B.), at which the question had been raised by G. Y. Zinoviev, one of the leaders of the “new opposition”, that Stalin had offered his apologies.

https://www.marxists.org/archive/lenin/works/1923/mar/05.htm

Here is something you can read about the so-called "Lenin's testament" forgery:

https://mltheory.wordpress.com/2017/02/07/on-the-alleged-forgery-of-lenins-testament/

https://www.revolutionarydemocracy.org/rdv7n1/LenTest.htm

And here is a debunking of the alleged Bukharin and Tito "found letters" story, in which the Lenin letter about Krupskaya is also mentioned:

https://msuweb.montclair.edu/~furrg/research/bukhlastplea.html