How do I validate TextFields in an UIAlertController?

This can be done by extending UIAlertViewController:

extension UIAlertController {

    func isValidEmail(_ email: String) -> Bool {
        return email.characters.count > 0 && NSPredicate(format: "self matches %@", "[a-zA-Z0-9._%+-]+@[a-zA-Z0-9.-]+\\.[a-zA-Z]{2,64}").evaluate(with: email)
    }

    func isValidPassword(_ password: String) -> Bool {
        return password.characters.count > 4 && password.rangeOfCharacter(from: .whitespacesAndNewlines) == nil
    }

    func textDidChangeInLoginAlert() {
        if let email = textFields?[0].text,
            let password = textFields?[1].text,
            let action = actions.last {
            action.isEnabled = isValidEmail(email) && isValidPassword(password)
        }
    }
}

// ViewController
override func viewDidLoad() {
    super.viewDidLoad()

    let alert = UIAlertController(title: "Please Log In", message: nil, preferredStyle: .alert)

    alert.addTextField {
        $0.placeholder = "Email"
        $0.addTarget(alert, action: #selector(alert.textDidChangeInLoginAlert), for: .editingChanged)
    }

    alert.addTextField {
        $0.placeholder = "Password"
        $0.isSecureTextEntry = true
        $0.addTarget(alert, action: #selector(alert. textDidChangeInLoginAlert), for: .editingChanged)
    }

    alert.addAction(UIAlertAction(title: "Cancel", style: .cancel))

    let loginAction = UIAlertAction(title: "Submit", style: .default) { [unowned self] _ in
        guard let email = alert.textFields?[0].text,
            let password = alert.textFields?[1].text
            else { return } // Should never happen

        // Perform login action
    }

    loginAction.isEnabled = false
    alert.addAction(loginAction)
    present(alert, animated: true)
}

Swift 4.0 Example

This is based on Mihael Isaev's answer. I had to change it up a bit to get the Save button to NOT be active immediately. I tried with and without the placeholder text. In the end, had to specifically inactivate Save to start with. In my case, I elected to use an alert title rather than placeholder text. But, it worked the same either way.

let alert = UIAlertController(title: "Enter Username", message: nil, preferredStyle: .alert)

alert.addAction(UIAlertAction(title: "Cancel", style: .cancel, handler: { (action) -> Void in}))
let saveAction = UIAlertAction(title:"Save", style: .destructive, handler: { (action) -> Void in
})
alert.addAction(saveAction)
alert.addTextField(configurationHandler: { (textField) in
    textField.text = ""
    saveAction.isEnabled = false
    NotificationCenter.default.addObserver(forName: NSNotification.Name.UITextFieldTextDidChange, object: textField, queue: OperationQueue.main) { (notification) in
        saveAction.isEnabled = textField.text!.length > 0
    }
})
self.present(alert, animated: true, completion: nil)

Most elegant way is to use

NotificationCenter.default.addObserver(forName: NSNotification.Name.UITextFieldTextDidChange...

Swift 3.0 example

let alert = UIAlertController(title: nil, message: nil, preferredStyle: .alert)
    alert.addAction(UIAlertAction(title: "Cancel", style: .cancel, handler: nil))
    let saveAction = UIAlertAction(title:"Save", style: .destructive, handler: { (action) -> Void in

    })
    alert.addAction(saveAction)
    alert.addTextField(configurationHandler: { (textField) in
        textField.placeholder = "Enter something"
        NotificationCenter.default.addObserver(forName: NSNotification.Name.UITextFieldTextDidChange, object: textField, queue: OperationQueue.main) { (notification) in
            saveAction.isEnabled = textField.text!.length > 0
        }
    })
    present(alert, animated: true, completion: nil)