The phrase is generally attributed to (president) Ronald Reagan by way of a Russian proverb. Reagan may well have popularized the expression (in English) but it appears in print somewhat earlier; for example, translated from a "Khrushchevian motto" in the Jan/Feb 1966 issue of Problems of Communism. I'm going to suggest prior usage much further back in time, albeit significantly out of context. The following quotation negates the 'trust' part, but doesn't that make more sense?
"The secret of success in the practical study of facts is to observe carefully and patiently; to take nothing on trust, but verify the truth of every appearance; to test the accuracy of every impression, and bring each inference to the trial of fact without daring to make any generalisation."
[The Lancet: To the medical student. London, 13 Sep 1879]