If I go on a tour at Barnard, will they check if I apply to Barnard Bound?
I’m unclear as to what your ultimate question is. Barnard Bound and campus tours are not related? You can do both, either (or none), as they serve different purposes.
Your regular college tours will give you a first-hand look at the colleges you are considering, so that can narrow a choice to a few that feel like the best match.
“Barnard Bound” is a program for students of color that affords them the opportunity to spend three days in New York for free, getting a feeling for the neighborhood, the student body, and of dorm-life in general at Barnard. One of the days is the general “Open House”.
You probably wouldn’t need another college tour AFTER the “Open House” (until hopefully the “Accepted Student” visit), but it’s entirely possible that you had previously toured Barnard at some point and this year decide to apply for “Barnard Bound”.