As you've discovered, TextInput is for a single line of editable text.
The right component to use for multi-line editable text depends on your needs. On desktop, you should definitely use TextArea. On mobile, you can use TextArea too, but you may find that TextInput is a better choice on mobile if it uses a StageTextTextEditor with the multiline property set to true. StageTextTextEditor is based on flash.text.StageText, which often provides the best user experience on mobile, including native copy/paste, scrolling, etc.
var input:TextInput = new TextInput();
input.textEditorFactory = function():ITextEditor
var textEditor:StageTextTextEditor = new StageTextTextEditor();
textEditor.multiline = true;
this.addChild( input );