I suggest you have a look at jQueryUI: http://jqueryui.com/resizable/#textarea
They have a lot of well documented widgets, the resizable looks like something that meets your demands. There are a lot of customization attributes, so you'd have to look into the documentation to see which ones you have to use
let me begin by pointing out that dragging the edge of a text area to increase its size is an out of box apex feature
how you can ccreate a textarea item and then use the following code to achieve the necessary effect
put this code in the Header Text text area of Header and Footer section of page defination.
here P3_X is my textarea item. i have uploaded jquery.autosize.min.js and associated it with the workspace. you can get this file from http://www.jacklmoore.com/autosize/
let me know if this answers your query in the current thread