How can I make a button have a rounded border in Swift?

Use button.layer.cornerRadius, button.layer.borderColor and button.layer.borderWidth.
Note that borderColor requires a CGColor, so you could say (Swift 3/4):

button.backgroundColor = .clear
button.layer.cornerRadius = 5
button.layer.borderWidth = 1
button.layer.borderColor = UIColor.black.cgColor