For example referring to an extremely cautious woman as a belt-and-suspenders kind of woman. Is it correct English (Am. English)?
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I believe that suspenders when worn by men are what are called braces in British English. Hence we have the expression belt and braces, applied to a course of action in which the participants wish to be doubly sure. Maybe this is what you have heard. It is not necesssarily applied to either a man or a woman.
In days when suspenders (BE:braces) were standard attire for men, wearing both a belt and suspenders was a common expression for excessive caution: making assurance doubly sure, so to speak.
But women (except the young-and-hip, especially in the late 1970s, when Annie Hall* was popular) have rarely worn suspenders. And as the comments to the question tell you (and as you probably realized), the phrase suggests garterbelt/suspender belt when applied to a woman—only mildly salacious, perhaps, but Not Suitable For Work, as it may contribute to creating an actionably hostile work environment.
So what can you use instead? I don't know of an established catchphrase, but I offer this analogy as a stopgap:
Belt and suspenders suggests not "getting caught with your pants down". Once upon a time the feminine equivalent would have been "getting caught with your slip showing", but that's pretty obsolete by now. A more contemporary indiscretion is jocularly labeled a wardrobe malfunction.
So one who takes redundant measures to avoid such embarrassing exposure might be described as a tape-and-safety-pin kind of woman.