I hate it when my ribbon frays, but I've found a few ways to solve this problem. Try any of them and let me know how you liked it.

1. If the end won't show on your finished project you can just use a piece of tape to cover the end of the ribbon.
2. You can Melt the ends of your ribbon. Melting simply involves a match or a lighter. Carefully hold the flame far from your ribbon while still being able to melt the edge. You can always melt a little more so it is better to go slowly. Also make sure to do this in a well ventilated area as the melting can give off a strong odor and also do this above your sink in case you burn yourself and accidentally drop the ribbon or lighter/match. Remember, you do this at your own risk, try not to burn the house down for this. :)
3. If you would prefer to fix the ends you can dab the ends of the ribbon with glue. Use a glue which will dry clear and is washable if you are using it on clothing. The only downside to this method is waiting for the glue to dry completely which usually takes several minutes to a few hours depending on the glue.