Try the following, not sure which will work for all browsers or the browser you are working with, but it would be best to try all:
<textarea style="overflow:auto"></textarea>
Or
<textarea style="overflow:hidden"></textarea>
…As suggested above
You can also try adding this, I never used it before, just saw it posted on a site today:
<textarea style="resize:none"></textarea>
This last option would remove the ability to resize the textarea. You can find more information on the CSS resize property here