The strategy of using Turnitin.com (or MyDropBox.com, for that matter) as a tool to make source use visible is a good idea. One of the truths we know is that sometimes students do in fact lose track of what words are theirs and what words come from sources, especially when they're moving beyond the more novice step of simply dropping in long block quotes and they try to find their own voice, and thus use more summary and paraphrasing.
There's a difficult transition both in the mechanics of paper production --the principles and guides for when, why, how and where to place quotes, cites, and signal phrases-- and in the intellectual work of merging voices --the writers' and the sources they're using-- smoothly and coherently.
So a technology that helps writers see where text in their essays possibly match text in sources is useful.
As much as I don't care for Turnitin's practice of hoarding student work in their database, their marketing copy that emphasizes detection over teaching, and their overstated promise to solve plagiarism, it cheers me that people are finding ways to make Turnitin.com useful if they're stuck with it. I'm not a fan of the program, and when I teach, I have students use CopyChecker, a small client side program that can be downloaded to their computer (When last I taught, it was free to students and may still be.).
With this, students can paste in one window their draft and in another window text from a source they are using. The program highlights matching and then students have a list of heuristics I give them: Is the match in your draft in need of quotations? Has it been cited? Should it be blockquoted?
We do an exercise as a class where they sticky-note/tab their handbooks so they have markers in the section of the book that will give them advice on answering those questions.
I require my students to have digital copies of any digital source they are citing. So for each source, they are required to save and download a copy (or copy and paste into the notes section of http://bedfordstmartins.com/bibliographer, a free for any student/teacher to use tool I like [reminder: I work for Bedford/St. Martin's] better than citationmachine and other ad-supported tools of that kind). Because students have copies of their source material, CopyCatch is great tool. It makes checking for matching text a part of their research drafting process.
However, in a writing center, where you don't have copies of CopyCatch, or if you did, you're likely not always going to have a student arrive with his or her original source in digital format, a WWW based search and match and compare tool is the only way to do this kind of thing.
If your campus has a Turnitin.com and/or MyDropBox license anyway, one of the ways you can lead and teach good practice instead of police-state/drug-test uses of the program, is to proclaim the pedagogy and service you're providing. Advertise to faculty. Present it at faculty orientations so they see a model of better pedagogical use. Walk them through a use case, showing how a paper is uploaded, matches are found, and how a tutor helps the writer ask the right questions, shows them how to open a handbook for the right advice and guidelines, and helps them to learn from mistakes, helps them to learn revision, helps them to find their voices, and helps them to better frame an argument.
Because once you find these mis-uses of a source, you have a really good teachable moment that goes miles beyond the mechanics of where to place a quote and a cite. A demo of how you use it before faculty whenever you get the chance to advertise your center and to paint a pedagogical picture others can follow.