You don't need any specific degree to be a fashion editor. Anna Wintour never went to university, and Andre Leon Talley studied French Literature at Brown. However, the former was born into publishing and ALT was in with Warhol and Diana Vreeland among others.
Honestly, any job in fashion journalism is going to be tough if you aren't already really wealthy and/or well-connected. Most editorial assistants in New York make around $18-22,000 a year, which is practically impossible to live on in the city. AW picks up most of her interns and assistants at Barnard and isn't particularly nice to them either.
None of this should stop you from pursuing your dream, though. A great degree in journalism from maybe Columbia or the London College of Fashion coupled with some impressive internships would be a great start.