I've implemented this with:

extension String {
func hasPrefix(other:String) -> Bool {
let otherCharacters = other.characters
if otherCharacters.isEmpty {
return false
}
let otherStart = otherCharacters.startIndex
let otherEnd = otherCharacters.endIndex

let selfCharacters = self.characters
let selfStart = selfCharacters.startIndex
let selfEnd = selfCharacters.endIndex

var s = selfStart
var o = otherStart

while ( s < selfEnd && o < otherEnd ) {
if selfCharacters[s] != otherCharacters[o] {
return false
}
s = s.successor()
o = o.successor()
}
return o == otherEnd
}
func hasSuffix(other:String) -> Bool {
let otherCharacters = other.characters
if otherCharacters.isEmpty {
return false
}
let otherStart = otherCharacters.startIndex
let otherEnd = otherCharacters.endIndex

let selfCharacters = self.characters
if selfCharacters.isEmpty {
return false
}
let selfStart = selfCharacters.startIndex
let selfEnd = selfCharacters.endIndex

var s = selfEnd
var o = otherEnd

repeat {
s = s.predecessor()
o = o.predecessor()
if selfCharacters[s] != otherCharacters[o] {
return false
}
} while ( s > selfStart && o > otherStart )
return o == otherStart
}
}

Which file should it go into? The existing hasPrefix/hasSuffix on OSX in stdlib/public/core/StringLegacy.swift but I don't know if that's the right place to put it.

Other potential pulls implementing the same features:

#440
#630

Created pull request for the above at #737

Fixed in f9b3e14.