For a small percentage of users, badges are extremely motivating. As a case study, consider the introduction of the Reviewer badge. The badge was first awarded on Nov. 7, 2011, but it was not announced until Nov. 20. Here's the number of reviews per week on Stack Overflow:
The Strunk & White badge is a prerequisite for getting Reviewer and here's the number of people who were awarded each over the same period:
The numbers practically speak for themselves: a small number of users were suddenly motivated edit in order to get S&W so that they could have an opportunity to earn Reviewer. Review tasks increased noticeably after the badge was announced. The evidence strongly suggests that if you want to increase certain behavior, a badge is a fine place to start.
It also stands to reason that once someone earns a one-time badge, they will no longer be motivated to do whatever it is the badge encourages. I don't currently have the data to support that claim. I would want to look at a badge, such as Copy Editor, that is awarded just once and check how much the rate of the triggering behavior slows after individuals get the badge.
On the other hand, we might not be much concerned if people don't continue to do whatever it was they did to earn the badge. Certainly, all other factors equal, we'd like to have people do more of the things we encourage with badges. But if some (or even most) people lose interest in doing a particular task after the badge is earned and enough new people arrive to do what needs to be done, there's no particular reason to be concerned.