For a quick clean, use a piece of moistened cotton to wipe the strap and wipe carefully.
If thorough cleaning of your watch is required, you could always use toothpaste. Yes, you read that right. Just use a minimal amount of mild toothpaste and a soft toothbrush to clean the mesh straps. Lightly dampen a piece of cotton with water to remove all the toothpaste once you've lightly brushed it. Unless you like the fresh smell? We're kidding, wipe the toothpaste off after, that's weird.
If you have any doubt about cleaning your watch, you should contact your local watchmaker or watch retailer for cleaning products specifically designed for mesh.