Original proof that "almost all decision problems are uncomputable"?
So the answer to my original question is the following: Because the statement is a corollary of the two facts that 


every program (Turing machine) is a finite binary string, i.e., a natural number,
every decision problem is a function $f: \mathbb{N} \to \{0,1\}$ and hence corresponds to a number $x \in [0,1)$ (and every such $x$ indeed encodes a meaningful decision problem, see Sasho Nikolov's first comment above)


the question about the original author / paper of this simple corollary is not that much interesting. (And the origins of the mentioned facts themselves I can trace back, of course.)

EDIT: I add here that this is the answer modulo the objection by Damiano Mazza (see in the comments above), which I need to check/think about first.