Should high schools pay for the cost of remedial education of college students? No.
First, not everyone should even go to college. It's not for everyone, and we shouldn't pretend that everyone should or could go to college. Believing that everyone should sends the wrong message to academically lower-performing students that they're failures. Yes, our schools should turn out students with at least a minimum proficiency level, but that's the opposite end of expecting everyone to go to college.
Second, imagine the untenable position high school teachers would be put in if the above suggestion were implemented. On the one hand we're already pressured to water down courses, either so graduation rates improve or because students today can't handle the work their peers of 10 years ago could handle, but on the other hand we'd be chastised for sending too many unprepared kids to college. Lose-lose.
My solution is much more reasonable and puts the responsibility mostly on the student, where it rightly belongs. No remedial classes should be taught at universities. Remedial classes should be taught at community colleges which, in California, are still a bargain (although not as much of one as 30 years ago); students had one "free" chance to learn, now they pay for it. This would have the added beneficial effect of reducing university populations to such a degree that programs would no longer be impacted, and students would be able to get the classes they want/need and would be able to graduate on time.