Drop a cookie with a very long expiry date. It's not a 100% solution, as people can erase cookies, and of course use different devices to visit the same websites. That's the nature of the web.
If uniqueness is that important, then you should include an ID field in the survey, to identify each respondent. Users will obtain an ID by logging in or by e-mail, for example. You could then store the information you gather from the survey in a database.