Senate Hearings before the Committee on Appropriations, HR 17023, 1969. W. J. Driver, Administrator of the Veterans Administration.
Senator Magnuson: I will give you a good example. A disabled veteran I know thought he would like to be in the optical business. To be an optician, you would have to go to a university. He only wanted to be a technician working in an optician's office. There were quite a few thinking the same, so finally, a group of the eye doctors set up a school for them. Now, this is a profession that very few people think about when it comes to the disabled. However it fits right into a disabled person's abilities to make eye examinations or do technical work on visual corrections. I suppose the same would be true in dental laboratories and many other things of this nature.
Mr. Driver: Watchmakers, watch repairing.
Senator Magnuson: Yes.
Mr. Driver: You come down to the benefit program ....
Senator Allott: But excuse me, Mr. Driver, I don t have my correspondence here on this but on this same subject. I am just as interested as you and the chairman or anyone else is in seeing that we don t get into the situation that we were in after World War II where we had universities setting up courses in underwater basket weaving, and all this sort of thing.
In June 1944, President Franklin Roosevelt signed the Servicemen’s Readjustment Act of 1944 into law. Better known as the "GI Bill," the Act included the funding for multiple programs for veterans. Among them was funding for continued education for the veterans: "Up to $500 per ordinary school year for tuition, books, etc., paid to any accredited institution person chooses." It's outside the scope of this particular question to explore how and why Black, Mexican American and other veterans were excluded from the Bill but what's relevant here is the idea of "accredited."
Due to the courts and lawmakers interpretations of the 10th Amendment, there is no American education system to speak of - rather there are 50+ different systems, each with their own system of accreditation for K-12 and higher education. While a number of accreditation systems were around at the time the bill was passed, they were fairly new and not necessarily well-organized. State education departments did their best but accreditors were known to use reputation and word of mouth to determine which schools were accredited, rather than a careful look at course syllabus or pedagogical practice. This rubber stamp via reputation strategy was a familiar one as prior to World War I, higher education was primarily for the elite of America; which is to say, students were overwhelming white, non-disabled, Christian, and if not directly connected to power and money through family ties, they had social ties.
Slowly, through a series of battles and wars, the nature of higher education in America began to change. Colleges increasingly opened their doors to women and people of color, universities shook out into different colleges and colleges began to focus on topics of study, i.e. majors. This included closing "normal" schools an folding teacher-preparation programs into colleges but it also meant expanding career-specific trade or professional schools. Some of these schools were not unlike for-profit schools of the modern era - they opened quickly, got a rubber stamp accreditation, charged tuition and then closed. At the same time, there were cases where well-established colleges and universities offered courses aimed to students that were less rigorous than other courses they'd previously offered. (In some cases, these courses were about acclimating the students to college life or offering more vocationally-heavy content.) In effect, higher education became less elite.
So, that's the context around higher education in mid-century America. Basically, the 1950s and 1960s was a period of massive growth in American secondary and higher education and people needed new words and terms to describe what was happening. Why the particular turn of phrase "underwater basket weaving?" There's likely a couple of different reasons. First, it's difficult to say for sure who used the phrase first in the same way we can't really source the exact alpha origin of dropout or teenager. People in and around education needed a way to describe a course or program they felt was frivolous and "underwater basket weaving" fit the bill. Second, there's something aesthetically pleasing about the combination of sounds in the phrase. (Linguistics call this trait euphony, the opposite of cacophony, which describes combination of sounds that sound unpleasant or harsh.) which makes it easy to remember and easy to say. Third, it's a fairly evocative phrase. As the author of this decent write-up (which includes instances of colleges offering actual "underwater basket weaving" classes) noted, the phrase works in multiple ways. You may picture someone underwater weaving a basket, or someone holding a basket underwater while weaving.
(I'm not 100% sold that the phrase came from a mention in a stamp collector's magazine in a story about basket making practices by a group of Inuit people. Words and phrases in education generally spread through educational journals, books, and conferences and it's possible a stamp collector dropped it in reference to a course they thought was "easy" and it spread but I'm meh on that origin story. I appreciate the light touch the Wikipedia article authors give it.)