To open a date picker in textFieldShouldBeginEditing we can use the inpuView property of the textField.
To open a datePicker when text Field start editing we can follow these steps.

1- On your View Controller add a view that contains the our UIDatePicker and two buttons(Cancel and Done).
2- Also add two textField one will open the normal keyBoard and other textField will open the datePicker.
3- Create the IBOutlet for the datePickerSuperView,datePicker,keyTextField,dateTextField.
4- Connect these for outlet .
5- Set the textField delegate methods.

class ViewController: UIViewController, UITextFieldDelegate {
    @IBOutlet weak var datePickerSuperview: UIView!
    @IBOutlet weak var datePicker: UIDatePicker!
    @IBOutlet weak var keyTextField: UITextField!
    @IBOutlet weak var dateTextField: UITextField!
    
    override func viewDidLoad() {
        super.viewDidLoad()
        datePickerSuperview.removeFromSuperview()
        datePickerSuperview.hidden = true
        dateTextField.inputView = datePickerSuperview
        
        
    }
    
    override func didReceiveMemoryWarning() {
        super.didReceiveMemoryWarning()
        // Dispose of any resources that can be recreated.
    }
    
    
    @IBAction func btnDoneTapped(sender:UIButton){
        let dateFormatter = NSDateFormatter()
        dateFormatter.dateFormat = "dd/MM/yyyy"
        dateTextField.text = dateFormatter.stringFromDate(datePicker.date) as String
        self.view.endEditing(true)
    }
    
    @IBAction func btnCancelTapped(sender:UIButton){
        datePickerSuperview.hidden = true
        dateTextField.text = ""
        self.view.endEditing(true)
    }
    
    //MARK : TextField Delegate
    
    
    func textFieldShouldReturn(textField: UITextField) -> Bool // called when 'return' key pressed. return NO to ignore.
    {
        datePickerSuperview.hidden = true
        self.view.endEditing(true);
        return true;
    }
    func textFieldShouldBeginEditing(textField: UITextField) -> Bool {
        
        if textField == dateTextField  {
            
            datePickerSuperview.hidden = false
        }
        
        return true
    }
    
    
}