How do I scroll the UIScrollView when the keyboard appears?

The recommended way from Apple is to change the contentInset of the UIScrollView. It is a very elegant solution, because you do not have to mess with the contentSize.
Following code is copied from the Keyboard Programming Guide, where the handling of this issue is explained. You should have a look into it.

// Call this method somewhere in your view controller setup code.
- (void)registerForKeyboardNotifications
{
    [[NSNotificationCenter defaultCenter] addObserver:self
            selector:@selector(keyboardWasShown:)
            name:UIKeyboardDidShowNotification object:nil];
   [[NSNotificationCenter defaultCenter] addObserver:self
             selector:@selector(keyboardWillBeHidden:)
             name:UIKeyboardWillHideNotification object:nil];
}

// Called when the UIKeyboardDidShowNotification is sent.
- (void)keyboardWasShown:(NSNotification*)aNotification
{
    NSDictionary* info = [aNotification userInfo];
    CGSize kbSize = [[info objectForKey:UIKeyboardFrameBeginUserInfoKey] CGRectValue].size;
    UIEdgeInsets contentInsets = UIEdgeInsetsMake(0.0, 0.0, kbSize.height, 0.0);
    scrollView.contentInset = contentInsets;
    scrollView.scrollIndicatorInsets = contentInsets;

    // If active text field is hidden by keyboard, scroll it so it's visible
    // Your application might not need or want this behavior.
    CGRect aRect = self.view.frame;
    aRect.size.height -= kbSize.height;
    if (!CGRectContainsPoint(aRect, activeField.frame.origin) ) {
        CGPoint scrollPoint = CGPointMake(0.0, activeField.frame.origin.y-kbSize.height);
        [scrollView setContentOffset:scrollPoint animated:YES];
    }
}

// Called when the UIKeyboardWillHideNotification is sent    
- (void)keyboardWillBeHidden:(NSNotification*)aNotification
{
    UIEdgeInsets contentInsets = UIEdgeInsetsZero;
    scrollView.contentInset = contentInsets;
    scrollView.scrollIndicatorInsets = contentInsets;
}

Swift version:

func registerForKeyboardNotifications() {
    NotificationCenter.default.addObserver(self, selector: #selector(onKeyboardAppear(_:)), name: NSNotification.Name.UIKeyboardDidShow, object: nil)
    NotificationCenter.default.addObserver(self, selector: #selector(onKeyboardDisappear(_:)), name: NSNotification.Name.UIKeyboardDidHide, object: nil)
}

// Don't forget to unregister when done
deinit {
    NotificationCenter.default.removeObserver(self, name: NSNotification.Name.UIKeyboardDidShow, object: nil)
    NotificationCenter.default.removeObserver(self, name: NSNotification.Name.UIKeyboardDidHide, object: nil)
}

@objc func onKeyboardAppear(_ notification: NSNotification) {
    let info = notification.userInfo!
    let rect: CGRect = info[UIKeyboardFrameBeginUserInfoKey] as! CGRect
    let kbSize = rect.size

    let insets = UIEdgeInsetsMake(0, 0, kbSize.height, 0)
    scrollView.contentInset = insets
    scrollView.scrollIndicatorInsets = insets

    // If active text field is hidden by keyboard, scroll it so it's visible
    // Your application might not need or want this behavior.
    var aRect = self.view.frame;
    aRect.size.height -= kbSize.height;

    let activeField: UITextField? = [addressTextView, servicePathTextView, usernameTextView, passwordTextView].first { $0.isFirstResponder }
    if let activeField = activeField {
        if !aRect.contains(activeField.frame.origin) {
            let scrollPoint = CGPoint(x: 0, y: activeField.frame.origin.y-kbSize.height)
            scrollView.setContentOffset(scrollPoint, animated: true)
        }
    }
}

@objc func onKeyboardDisappear(_ notification: NSNotification) {
    scrollView.contentInset = UIEdgeInsets.zero
    scrollView.scrollIndicatorInsets = UIEdgeInsets.zero
}