Allowing citizens to "vote" in opposition to a specific petition will help to channel and encourage civic engagement in the "We the People" Petition service.
Such action may help to focus the petitions listed here. It may help to reduce petitions which are based in hatred toward other groups. It could help to alert citizens that a petition may be rooted in poorly-informed opinions.
And it will help the White House to recognize, with a measure of certainty, when a petition is regarded as spurious, or unwanted, by most Americans.
"Thumbs down" votes may not have any binding impact, but the opportunity to speak out, for or against a given petition, would be a valuable contribution in itself.