DAYTON, Ohio — When Hillary Clinton was searching for a running mate, she made clear her top criterion: Her selection needed to be ready to become president “if something were to happen,” as she put it.

When she announced Senator Tim Kaine of Virginia as her choice, Mrs. Clinton affirmed that he had passed that test.

Now, after Mrs. Clinton had to be helped into a van while departing a Sept. 11 anniversary ceremony on Sunday and the later disclosure that she had been given a pneumonia diagnosis on Friday, Mr. Kaine is coming under a more intense spotlight. When he took questions from reporters on Monday, cable news channels carried the exchange live, a rare level of attention for his candidacy.

While everyone — politicians included — gets sick from time to time, the intense focus on Mrs. Clinton’s health brings with it a constitutional reminder: Mr. Kaine is not only a campaign-trail booster, but also a possible president himself.