I believe that this derives, via a game of whispers, from a line in The Idiot, where Ippolit Terentyev says:
— Нет, а за то, что недостоин своего страдания.
Fyodor Dostoevsky (1869). The Idiot, part 4, chapter 5. Wikisource.
Here are three translations into English:
None of these is very close to the version quoted in the question. But let’s trace Frankl’s version of the quotation backwards. The first part of Man’s Search for Meaning (1959) is an English translation, by Ilse Lasch, of Ein Psycholog erlebt das Konzentrationslager (1946), which says:
Dostojewski hat einmal gesagt: »Ich fürchte nur eines: meiner Qual nicht würdig zu sein.«
Viktor E. Frankl (1946). Ein Psycholog erlebt das Konzentrationslager. Vienna: Verlag für Jugend und Volk. (Thanks to svavil in comments for locating and transcribing this.)
This does not correspond closely to the published German translations of The Idiot:
||E. K. Rahlin
||„Nein, deshalb, weil ich dieses Leidens unwürdig bin.“
||»Nein, weil ich meines Leidens nicht würdig bin.«
However, Frankl’s version of the line is very close to the version given by Georg Brandes in his essay on Dostoevsky:
Eine Person bei Dostojewski sagt:† „Ich fürchte nur, meiner Qual nicht würdig zu sein.“
A person in Dostoevsky says, “I just fear not being worthy of my suffering.”
Georg Brandes (1900). ‘Fyodor Dostojewski’. In Menschen und Werke, p. 343. Frankfurt: Rütten & Leoning.
† Note the similarly with Frankl’s “Dostojewski hat einmal gesagt”: Brandes did not say who the speaker was, and so Frankl could not name him either.
Frankl’s version is identical to Brandes’ version except for the addition of “eines” after “nur”, likely in an attempt to disambiguate the wording. But where did Brandes get his translation from? A plausible source is a monograph on the Russian novel by Eugène-Melchior de Vogüé, who quoted the line in French translation:
Ailleurs il dira:† «Je crains de n’être pas digne de ma souffrance.» Et cent autres semblables.
In another place he says: “I am afraid I am not accounted worthy of my sufferings.” There are hundreds of other sayings all in the same strain.
Eugène-Melchior de Vogüé (1886). Le roman russe, p. 259. Paris: Plon. Translated by H. A. Sawyer (1913). The Russian Novel, p. 253. London: Chapman and Hall.
† De Vogüé does not name the character, and the last name mentioned was Myshkin on page 258, so if Brandes had been relying on Le roman russe, it would have been tricky to figure out who was speaking in the quoted line, and so he might have written “Eine Person” to avoid getting it wrong.
So, in Russian, the line had the temporizing phrase, “а за то” (but for that). In de Vogüé’s translation, this became “je crains” (I fear), a conventional piece of politeness, not an actual expression of fear. In Brandes this became “Ich fürchte nur” (I just fear), the phrasing of which was ambiguous between conventional politeness and literal fear. And in Frankl this became “Ich fürchte nur eines” (I fear only one thing), resolving the ambiguity in the wrong, but more dramatic and interesting, direction.