[1] https://wwws.whitehouse.gov/petitions/%21/petition/actually-...
The petitions are not there for us to influence them. They're there (they spent effort to put them there) for them to influence us.
I would view it as one more avenue of making your opinion known to elected officials. In addition to informing your congressperson and senator to try to prevent the bill from passing this would be a way to endorse a veto from the executive branch.
I believe it does as much good as anything else you can do in the amount of time it takes to sign it.
The problem being that most of the petitions that get any significant number of signitures are predictable , populist stuff like "Britain should leave the EU" , "bring back hanging" and "legalise Marijuana" , basically stuff that if the government had any inclination to do they would have already done by now so really all it does is waste everybody's time.
While you're there, I'm also a fan of the petitions that the petitions be taken seriously.
For instance, my father was under the impression that his will was totally legal and everything was taken care of. His misinformation led him to look no further, and as a result, when he passed, things were messy because the will wasn't valid.