It's a known bug that occurs more often if you have an outdated Flash player. You can test your Flash version by visiting this page: www.glitch.com/help/test/
I have the latest version of flash and this happens to me. Usually I hit the CTRL button to stop walking. I also sometimes have problems with constant menu scroll on piggies as well and use CTRL to stop the scroll.
That isnt a bug. If you hit Alt with your arrow key, you will continue to walk in one direction. To stop walking in that direction just hit the arrow button of the direction you are walking. I had a dev one time explain to me that it was intentionally that way.