Other commenters are completely right to mention his concern for proofs and the "Cruelty of Really Teaching Computer Science", but the most BASIC-specific thing that he was associated with was criticism of the GOTO statement.
In original BASIC, the GOTO is a foundational mechanism and a majority of programs would have used it, sometimes extensively. Dijkstra thought for many reasons that this wasn't good style and didn't promote clear thinking. And yes, one consequence of that is that it would be harder to prove programs correct or just to reason about whether they were correct.
Programs that overuse GOTOs (or from the point of view of later structured programming and functional programming advocates, perhaps programs that use GOTOs at all) were stigmatized as "spaghetti code".
By the way, this concern is not just about aesthetics: some of the ideas that Dijkstra was advocating are arguably those that newer programming languages like Haskell and Rust can use to find bugs in code automatically at compile-time, or to make it harder to write certain bugs at all. The line between Dijkstra's advocacy and these techniques is complicated but I think there is a connection. So partly we might say that Dijkstra was not just concerned with how to make it easier for humans to think clearly about program correctness, but ultimately also about how to make it easier for computers to help humans automatically determine (parts of) program correctness. And it's true that the GOTO style complicates that task.
Kind of ironic that nowadays many people in our generation consider the newer generations to be lacking fundamental education because they never used GOTO based programming languages. I've talked to multiple people who lamented that young programmers have never done assembly or BASIC.
It’s helpful to have a mental model of how the computer works. I don’t know if it’s necessary that one have spent mountains of time building real software using a GOTO/jmp style, but having exposure to it would be nice, rather than hiding it away.
Jeff Dunteman’s assembly programming books included a “chapter 0” that I always loved, and which really stuck with me for how creatively they taught those topics.
I mean, CPUs do a bunch of work to make us believe they still operate just as a fast PDP-11, and I would wager that besides compiler experts that work on the backend parts of compilers, not many people have a real feel for modern hardware (obviously besides those that actually work on that given hardware).
So I'm not convinced that even those who think they know how it works know it actually.
Not entirely. GOTO can be pretty nice! And even the lack of structs probably has the advantage of helping to prepare you for today's world where column-major is back in style, for performance reasons.
Dijkstra thought of computer science as a subdomain of mathematics, and thought that hands-on experimentation with actual computers would mostly lead students astray. A program should all be worked out and proven correct before (optionally) feeding it to a computer, and testing and even more so debugging were abhorrent practices.
BASIC, on the other hand, is more aligned with what Seymour Papert later came to call "Constructionism": the student learns by experimentation.
Ironically, I grew up with limited access to computers, so I wrote many programs on paper first, including a FORTH implementation in assembly language I wrote over summer break with a typewriter, waiting for school to start again so I could actually test it hands on.
https://homepages.cwi.nl/~storm/teaching/reader/Dijkstra68.p...
In original BASIC, the GOTO is a foundational mechanism and a majority of programs would have used it, sometimes extensively. Dijkstra thought for many reasons that this wasn't good style and didn't promote clear thinking. And yes, one consequence of that is that it would be harder to prove programs correct or just to reason about whether they were correct.
Programs that overuse GOTOs (or from the point of view of later structured programming and functional programming advocates, perhaps programs that use GOTOs at all) were stigmatized as "spaghetti code".
https://en.wikipedia.org/wiki/Spaghetti_code
By the way, this concern is not just about aesthetics: some of the ideas that Dijkstra was advocating are arguably those that newer programming languages like Haskell and Rust can use to find bugs in code automatically at compile-time, or to make it harder to write certain bugs at all. The line between Dijkstra's advocacy and these techniques is complicated but I think there is a connection. So partly we might say that Dijkstra was not just concerned with how to make it easier for humans to think clearly about program correctness, but ultimately also about how to make it easier for computers to help humans automatically determine (parts of) program correctness. And it's true that the GOTO style complicates that task.