Because I'm old, I still participate in that old person's technology, the listserv. On one of these lists, the question of if/how technology supports cheating came up. I've seen this question in various forms over the years; in this specific instance, it came up in response to a pilot program with Google Docs.
There are lots of excellent reasons not to use Google docs in your school, but cheating isn't one of them.
As people were responding within the list, some people recommended using TurnItIn.com as a deterrent to cheating.
And given that using TurnItIn.com is really bad advice, I felt compelled to offer some alternatives.
Teach writing as a process
Teach writing as a process. If all you see from a student is a final draft, you will have a hard time knowing how that final draft came to be, and you will be less effective at helping a writer improve. If you teach writing as a process, and see pieces of work from initial conception (this is my thesis), through notes, through a first draft, a second (and subsequent) drafts, through to the "final" paper, you will be able to give more targeted feedback. Using a working portfolio (aka, a blog) is a great tool for teaching process.
Students are honest
Approach your subject from the perspective that your students are honest. I know, crazy talk here. But people will generally rise to the expectations you set for them. Nothing says "you are not worthy of trust" better than using a system like TurnItIn.
Know style, and teach style
Know style, and teach style. People should know how to spot (and when to use) active verbs and passive verbs. People should know that a simple technique like scanning a paper for overuse of "to be" verbs will do wonders for their sentence structure.
People should know the different sentence structures, and when a simple sentence is a better choice than a compound-complex sentence. They should know how to analyze their own writing for variability within sentence types, and the effects it has on pacing. They should be able to spot repetitive patterns within their paragraphs, and either fix it or use it to their best rhetorical advantage.
People should know to examine their word choice, and the advantages and disadvantages of using words that are latinate versus anglo-saxon in origin. They should know to look for average sentence length, average paragraph length, and the average word length within a representative section of their writing.
Every writer has a distinct style. When you begin looking at writing and analyzing style, words written on a page become as distinct as the sound of a person's voice.
Technology Does Not Have Agency
Making the claim that using Google Docs (or a word processor, or a typewriter, or a printing press, or a hired scribe) makes it more likely that students will cheat misses the point. You know who is doing work by talking with them about that work. The technology is a means to getting work done; imbuing it with the agency to support cheating is a profound misunderstanding of both technology, and of what motivates people to do their best work.
Using a system like turnitin.com is a great way to tell your students "I don't trust you, and I'm not willing to take the time to know how you think."
Cheating is not a technological issue. To minimize incidents of cheating:
Provide challenging, stimulating assignments;
Check and provide feedback on in-progress milestones;
Talk with your students;
Teach style; and
Be clear with your guidelines and your expectations. The more direct and clear you are with your students, the more direct and clear they will be with you.