You could mute the ringing open note with your picking hand. That is known as palm muting. By applying pressure on the ring note with your picking hand, you'll kill the ringing.
The other way I can think of would be muting using your fretting hand. Let's say you play the 4th string open and the next note is on the 5th string. You could, when playing the second note, put a little angle on your fretting finger so that you are touching the 4th string. Than would also kill the ringing.
Hopefully I'm explaining this clearly enough and it can help you out.