One could argue that the fans save any show that succeeds -- because they watch it in enough numbers that it gets good ratings. The most important factor is always going to be whether the network and the advertisers feel that the ratings are sufficient to offset the expense of making the show. The number of people who write letters in support of a show is generally going to be a fraction of the number of viewers it would take to ensure a show could break even. Maybe sometimes it can help assure a network that there is audience interest, or bring publicity to the show, but unless that pays off in actual ratings, it won't make much difference.
There have been a few cases where letter campaigns have convinced network execs to keep a show or even uncancel it, though usually in concert with other factors.
This article has a list of them, including the standard
Star Trek myth, but let's focus on the others. In the case of
Chuck, the deciding factor was probably the fans' efforts to convince Subway to come on board as a sponsor, which had an impact because there was actual money involved for the network rather than just enthusiasm.
Jericho's letter campaign worked because it convinced the network to review the ratings more carefully and learn that Nielsen had underreported them.
Friday Night Lights survived because the network was able to enter a cost-sharing arrangement.
So, yeah, sometimes it can make a difference, but rarely by itself. It's generally going to be some additional factor that makes the crucial difference. At the very least, there probably has to be someone at the network who wants the show renewed anyway and just needs a reason to be convinced they're making the right decision. But the show has to be enough on the bubble that it's at least credible that it could get better ratings.
In the case of
Trek, there's so much mythology around it that it's hard to say. Fan enthusiasm may have been a contributing factor, but there's no clear evidence that the show wouldn't have been renewed for a third season anyway, even without the campaign. It was on the bubble, but that's all we know for sure. NBC had good reason to want to keep the show. It was a smart show that brought prestige due to the award nominations it garnered. It was visually striking and encouraged people to buy RCA's color television sets. So they would've wanted a way to keep it. And they did what networks generally do to shows with sagging ratings in order to keep them around: They slashed the budget. The third season had fewer episodes, virtually no location work, smaller casts per episode, etc.
So no, I don't think the letter-writing campaign "saved the show," not as the single deciding factor. Maybe it was a contributing element, an affirmation that the show had a loyal audience, but it wasn't the only reason the network had to keep the show around.