You could try to use a barely moistened cotton ball or soft cotton material to clean your mesh strap. Gently rub it over the the straps, and use a dry cotton ball or cloth afterward to dry the straps.
If a more thorough cleaning of your watch is needed, you could try using a minimal amount of mild toothpaste and a soft toothbrush to clean the mesh straps. Lightly dampen a cotton ball to remove all the toothpaste.
Please be careful not to wet the watch itself.