There are remedial things you can consider - making sure you have hard-wearing grass species, decompacting the soil, improving the drainage and mowing with a higher setting.
If the erosion cannot be cured by maintenance practices alone and you want the paths to remain grassy, perhaps you should look at some form of reinforcement materials,which are intended to take the strain of traffic.
It may be that you could use stepping stones in the areas most at risk but if you want to keep the grass element, you should research meshes and geotextiles for incorporation into the surface.
While some meshes can be laid on the surface, with the grass growing through, installing others can mean major works but it can provide a long-term solution. It is essential to realise, though, that any underlying compaction and drainage problems must be sorted out first.