I can’t remember how, on a recent drive to New York, my husband and I got started on a discussion of the phrase strait and narrow. But I do know that, absent a dictionary to straighten things out for us (neither of us being fond of checking Google while roaming), we determined that both the traditional spelling and the more common contemporary phrase, straight and narrow, made no sense.
On the one hand, with strait meaning “narrow or cramped,” strait and narrow seems redundant. The phrase is most used for someone who has strayed from a conventional path and needs to use willpower to stick with the program. Certainly it makes sense to assume that, especially for someone used to a certain amount of lifestyle or moral latitude, such a path seems narrow. Why must it be strait also?
On the other hand, who decided that the correct path forward would be straight? Most often, it’s not. Metaphorically, it’s easy to picture someone stumbling straight on ahead, perhaps over a cliff, when they need to pay attention to this narrow path as it twists and turns.
When we got home, of course, I looked it up, and discovered that the origin is the Book of Matthew as translated in the King James Bible:
Enter ye in at the strait gate: for wide is the gate, and broad is the way, that leadeth to destruction, and many there be which go in thereat:
Because strait is the gate, and narrow is the way, which leadeth unto life, and few there be that find it.
There are two items here, a gate and a path, which our idiom collapses into one. Thus the apparent redundancy of strait and narrow. The more popular version today, having more than traded places with strait and narrow, as this Google Ngram shows, 
may have other roots. According to the Oxford English Dictionary, the use of straight to denote bold or steady dates back to 1540; to mean frank or honest, to 1530; to mean well-conducted, to 1853. The meaning of straight as conventional or respectable (and, by extension, heterosexual) is more recent – since 1941, per the OED — but it fits the pattern. The image that always comes to my mind when I hear straight used in these ways is of an individual with exemplary posture, the straightness being more vertical, if you will, than horizontal. Still, I expect it is the accumulation of these meanings that makes straight and narrow sound logical even if it’s untrue to the original sense. Tellingly, checking the Corpus of Contemporary American English, I find no matching strings at all for strait and narrow, whereas there are 226 for straight and narrow.
Strait, then, would seem to be on the ropes. It still wins the day, according to both Google and COCA, with dire straits and geographical terms like Straits of Gibraltar — in other words, where it’s used as a noun, since straight’s use as a noun is relatively rare. Straitlaced, on the other hand, is gradually giving way to straightlaced. Here again, there’s a modicum of sense to be found in the idea of laces (presumably for a corset) being pulled so tight that they not only narrow the torso of the poor soul within, but also come closer to a straight line than to a zig-zag pattern. And Merriam-Webster gives us a good explanation of the evolution of straitjacket into straightjacket.
I can’t find any logic in another malapropism, straightened circumstances rather than straitened circumstances; I’d welcome any suggestions on that one. What I do note is that for all these substitutions of straight for strait, the source is not some self-published romance novel, but venues like NPR, Academic Questions, The New York Times, and Newsweek. My prediction is that we’re on our way to an additional meaning for straight, in which it is a synonym for narrow or constricted. And the only shame of it is the insertion of yet another unnecessary gh into an otherwise perfectly good word.