How do I stop a creek from eroding my steep embankment?
This is often a complex legal situation due to regulations that affect (in many parts of the world) what you can do in or near a body or stream of water even if the land next to it or around it belongs to you; not to mention the fact that you are fighting with a force that has literally shaped the planet, and it's inexorable and quite capable of undoing many things you try to do to stop it, if, in fact, it is even legal for you to try to do those things. If the stream also is a property BOUNDARY, the legal implications of trying to move it get even more complex.

The "wire mesh" you describe is most probably gabions, which are empty wire cages which are stacked and filled (pretty much by hand, which has taken a novice contractor or two by surprise, to their cost) with rocks. Another common technique is "rip rap" which are simply large rocks layed over the face.

Your only legal option in many places would be to think about moving your fence back before it's undermined. Trying to get plant material established on the eroding slope is another approach, but many jurisdictions are less fond of people planting highly invasive species often selected in the past for that purpose, so you may need to stick with native/local species. Or, you may be able to perform some work after having hearings/meetings with the Authority Having Jurisdition over such activities in your area (conservation commission, department of environmental protection, etc.)