Unfortunately, the time delay between childhood events and students' career choices as adults means it's difficult to link choices to one event in their childhood. However, we can look at adults' actions to see if they thought the moon landing would have an impact on future career options for young people (by which they meant mostly boys, mostly white, and those without disabilities.)

During the colonial era and the first half of the 1800's, American students in school generally encountered a classical curriculum - which meant a focus on Latin, Greek, rhetoric, logic, math, and some sciences. This model privileged learning things for the sake of learning; smart men knew certain things so young people should study those things to be smart. Slowly, schools shifted to a more liberal arts curriculum (also known as a modern or English curriculum) which included more "practical" learning such as literature, modern languages like French, physical education, music, art, science. This transition happened at the same time as the rise of the common school and an understanding that American schools would focus on helping students be good citizens and good Americans, not necessarily focus on their choices after leaving school. (More on vocational education in schools.) This doesn't mean schools never focused on life after school but rather, there was a general sentiment that a student could grown up and be whatever he wanted to be. (Any maybe his sister. Possibly. Probably not Black, Hispanic, Asian, and Indigenous children.) In 1894, The NEA Committee of Ten surveyed the country and drafted recommendations about high school curriculum. The authors of the section on Science education stressed that students should get hands-on experience with phenomena, labs, and the natural world.

There was a further shift in the mid-1900's to sharpen the focus on math and sciences and to elevate them above content like art, music, and physical education. This tightening came about due to a number of variables - and it's safe to say the space race was a significant reason why. NASA started working closely with schools, colleges increased opportunities for dual enrollment in math and science classes, and teacher prep programs actively recruited more teachers for math and science positions.

To be sure, though, young people have been interested in math and sciences for centuries. This question gets at the history of female "computers" and their courses of study.