Except the whole "The letter-writing campaign saved Star Trek" thing is a myth. There's no proof that the show was ever actually slated for cancellation; it was just on the bubble. NBC wanted to keep the show, since it brought them some prestige (it got Emmy nominations every year) and it encouraged the sale of color televisions, which brought profit to their parent company RCA. They just had to investigate whether they could trim the budget enough to justify keeping it for another year, which is the case with many shows, and eventually they decided they could. So they would've most likely renewed it anyway. The announcement they made on TV telling fans that the show had been renewed wasn't really saying "We surrender, you've convinced us not to cancel it" -- it was just saying "Hey, we aren't actually cancelling the show, so please stop flooding our poor mailroom employees with all this unnecessary mail."
As a rule, letter-writing campaigns don't work, because goodwill doesn't pay for a show. Networks decide whether or not to produce shows based on whether they think they can make enough profit to justify the expense. Even a large letter-writing campaign rarely involves enough participants to convince a network that the audience would be large enough to make a show profitable, especially if it's an expensive SF show. And petitions are even less helpful. It takes very little effort to sign a petition, and the act doesn't guarantee that you'd actually commit to watching the show. I'm sure such displays of fan support are something the networks and studios take into account, but a petition or letter-writing campaign wouldn't be enough by itself to make the difference. The network would have to take other factors into account in assessing the feasibility of the project.
So a petition wouldn't hurt, but we shouldn't have unrealistic expectations about its likely impact.
