The present perfect is appropriate here because the sentence is really about the present. It matters to the university whether the applicant "has done a masters before" because, presumably, the applicant who has is deemed more competent and knowledgeable today by the university.
The simple past, as the name suggests, is used to describe a past occurrence with no intended implication about the present. For example: "My late uncle did his masters at NYU."
The rule erdk suggests is perhaps a useful shortcut for taking a test, but you should understand how the rule came to be. For the present perfect, since the emphasis is on the present, it usually doesn't make sense to mention the time. "I've seen the film before, so I don't want to see it now, (It doesn't matter whether I saw last week, last month, or last year. I don't want to see it today.) Note, though, that this isn't the case when the time is introduced by the word "since," so the rule isn't foolproof. ("I've been a member since 1989.")
The opposite isn't always true either. There are instances where the simple past is appropriate even when the time is not specified. For example: "How did you meet your wife?" "We met at a bookstore." Here, the subject matter is the past, so the past tense is appropriate, even though the time isn't mentioned.