Wikipedia does not actually know that it is from 1937, my guess is that they picked a number between 1929 and 1950. According to Lagarias, who compiled an exhaustive bibliography on the problem, the first publication only appears in 1963, and is by Klamkin, he also gives other names it acquired: Syracuse Problem, Hasse’s Algorithm, Kakutani’s Problem and Ulam’s Problem. Early history is addressed more specifically in his American Mathematical Monthly article, which cites first hand accounts (and links to some of them) that are not publications. Thwaites, who calls it Ulam's problem, recalled that he stated the $3x+1$ problem already back in 1952. Hasse promoted the problem during his visit to the Syracuse university in 1950s, Kakutani got involved around 1960:"For about a month everybody at Yale worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S."
Collatz himself circulated similar iteration problems at the International Congress of Mathematicians in 1950, although it is unclear if $3x+1$ specifically was among them. One such iterative function already appears in his notebook entry from July 1, 1932. Collatz wrote several letters in 1976-1980, where he reminisced about those early days, but as Lagarias points out "in none of these letters does L. Collatz actually state that he proposed the $3x+1$ problem".
He implies it though in a letter to Mays written in 1980 and translated here, and explains his motivation for studying such functions:"In 1929 I attended classes of E. Land and lectures in number theory given by Lettenmeyer, both in Göttingen, and in 1930 I attended classes given by O. Perron in Munich and Isai Schur in Berlin. I found it interesting to sketch the graphs of number theoretic functions $f(n)$ by drawing an arrow from $n$ to $f(n)$, or more simply, by writing $f(n)$ under $n$. (In this way) one can find various concepts which are well known in the theory of digraphs such as trees, cycles, bifurcation, etc. I don't know who the first person was to make these connections to graph theory; I, however, have seen neither in lectures nor in published form this type of representation of number theoretic functions. I enjoy observing the various patterns, and I computed the graphs (at the time of these investigations) of many interesting number theoretic functions for values of $n$ up to about $100$. I examined the example that I mention above [the conventional $3n+1$] in this way too..." This is as first hand and definitive as it gets without a contemporary document, and there is no 1937 in it.