It is not clear that the idea can be rigorously tested, in part because "complexity" can refer to at least three different things, and in part because there's a big sampling problem in checking if there is such a correlation. The worst possible way to try to answer the question is via cherry-picked languages evaluated subjectively. Chinese is obviously really simply morphologicially and it has a huge number of speakers, and Lushootseed is really complex morphologically, and it has very few speakers (none, in fact).
You could start by generating a random sample of let's say 100 languages, and check the facts, but I've engaged in that game before and it turns out that the biggest problem is getting an answer for most of those languages – in my most recent 100-language sample, only 5 languages have over a million speakers (Xiang Chinese, Southern Azerbaijani, Western Farsi, Sylheti, Georgian), and probably less that 20% of them have sufficient descriptive material that you have a prospect of saying how complex the language's morphology is. Since there are huge numbers of tiny languages and many well-populated dialects of Chinese, it's not clear to me what an unbiased sample would be.
One kind of complexity is "number of affixes", so if a language has just one affix slot and 200 affixes that could go there, that would be kind of complex, though mostly trivial. A second kind of complexity involves numbers of slots, so that most Indo-European languages would not be very complex since there are relatively few slots for affixes. The final and IMO best metric is in terms of the system of rules, for example if you get suppletive portmanteaux of slot1a+slot2d, or if there are morphological conditions on affixation (such as "add X only on perfective verbs"). While I think that gets at real complexity, I don't see how we can count (and if we can't count, we can't numerically correlate, and thus we're back to subjective judgments like "Turkish is complex" or "Kalaallisut is complex".
Still, I think it is probably true, based on subjective experience.