因为UITextView是UIScrollView的子类，它的bounds属性反映了坐标系的可见部分，所以应该可以这样做：

- (NSRange)visibleRangeOfTextView:(UITextView *)textView {
    CGRect bounds = textView.bounds;
    UITextPosition *start = [textView characterRangeAtPoint:bounds.origin].start;
    UITextPosition *end = [textView characterRangeAtPoint:CGPointMake(CGRectGetMaxX(bounds), CGRectGetMaxY(bounds))].end;
    return NSMakeRange([textView offsetFromPosition:textView.beginningOfDocument toPosition:start],
        [textView offsetFromPosition:start toPosition:end]);
}

这假设了一个从上到下，从左到右的文本布局。如果你想让它适用于其他布局方向，你将不得不更加努力。：）

Rob的答案，写在Swift 4中。我添加了一些安全检查。

private func visibleRangeOfTextView(textView: UITextView) -> NSRange {
    let bounds = textView.bounds
    let origin = CGPoint(x: 10, y: 10)
    guard let startCharacterRange = textView.characterRange(at: origin) else {
        return NSRange(location: 0, length: 0)
    }
    
    let startPosition = startCharacterRange.start
    let endPoint = CGPoint(x: bounds.maxX,
                           y: bounds.maxY)
    guard let endCharacterRange = textView.characterRange(at: endPoint) else {
        return NSRange(location: 0, length: 0)
    }

    let endPosition = endCharacterRange.end
    
    let startIndex = textView.offset(from: textView.beginningOfDocument, to: startPosition)
    let endIndex = textView.offset(from: startPosition, to: endPosition)
    return NSRange(location: startIndex, length: endIndex)
}

使用示例，通过点击按钮调用：

@IBAction func buttonTapped(sender: AnyObject) {
    let range = visibleRangeOfTextView(textView: self.textView)

    // Note: "as NSString" won't work correctly with Emoji and stuff,
    // see also: http://stackoverflow.com/a/24045156/1085556
    let nsText = self.textView.text as NSString
    let text = nsText.substring(with: range)
    NSLog("range: \(range), text = \(text)")
}