They'd be Will Smith in Independence Day, Amy Adams in Arrival, and that scrappy Henry Thomas kid from ET all rolled into one. The responsibility of first contact and managing relations with extraterrestrial life forms would fall on their shoulders, meaning they would have to be as diplomatic as Star Trek: The Next Generation's Jean-Luc Picard and as take-no-bullshit as Star Trek: Deep Space 9's Benjamin Sisko. It wouldn't hurt if they possessed the skills, fortitude, and constitution of Alien's Ellen Ripley, either.
So when it comes to the qualifications, they're not just hiring anybody here. And they don't want to have to fire them after less than two weeks on the job, so hopefully they'll pick someone halfway-useful.
In addition to the prerequisite background in physical science, engineering, or mathematics, the job description highlights three specific qualifications a great candidate should possess: 1. a deep understanding of the planetary protection responsibilities; 2. a bullet-proof record in "space programs of national significance"; and, finally, 3. a degree of diplomatic savvy "that resulted in win-win solutions during extremely difficult and complex multilateral discussions."