I think there's a balance. While a lot of the work being done in advanced mathematics may never impact anyone's lives in any substantial way that's not always true. It took a lot of effort to reinvent mathematics and geometry from the ground up in the 18th and 19th centuries, and in many ways it could seem as if that work was purely abstract, but it has surely had a very profound impact on the entire modern world. The same goes for, say, Boolean algebra, Turing machines (which at one time were incredibly abstract and theoretical), and a lot of number theory. Yet today our computing systems and especially cryptography is based on what used to be very abstract mathematics. Do you think the wonks who worked on elliptic curve theory imagined that there would eventually be custom purpose micro-chips designed to perform elliptic curve math?
Sometimes it's impossible to know what lines of research will prove practical in the next decades and centuries, so it's smart to balance between some focused practical application research as well as blue-sky research.
That's true. It's said that Turing himself didn't immediately realize the importance of his machines -- he went straight on to considering oracles and super computation, as if Turing machines themselves weren't worthy of further study.
Perhaps it is nit-picking, but I'm going to play devil's advocate and question exactly how important the theoretical notion of a Turing machine actually was for the practical development of computer algorithms. Probably a much more important development was the invention of the Von Neumann architecture and its use in the IAS machine and later the JOHHNIAC. And also lambda calculus for its spawning of functional programming languages.
Von Neumann is the best example I can think of a blue-sky thinker whose work had profound practical impacts all over the place. Maybe a recent example of an ingenious pure mathematician who made a practical impact is Terence Tao's invention of compressed sensing.
But on the whole, no-one would argue that the next cryptographic breakthrough is going to come from category theory. Some parts of pure mathematics will always be useless, and everyone knows it.
Sometimes it's impossible to know what lines of research will prove practical in the next decades and centuries, so it's smart to balance between some focused practical application research as well as blue-sky research.