The cracks are a phenomenon called crazing. They are typically caused by thermal stress. This is when the ceramic material is exposed repeatedly to hot and cold temperatures. This happens under the glaze of the basin and that's why it still feels smooth to the touch.
Because it's under the glaze there is no way to repair it, however, you can cover and recoat the basin withWhite Knight 500ml White Tub And Basin Paint. It dries to a tough, tile-like finish to restore previously discoloured surfaces.
Here is a handy step-by-step guide that you can follow:How to paint a bath. The technique used is applicable to painting a basin, too. Before painting, make sure that the surface is very well prepared so you get the best possible end result.
I also encourage you to have your doors and windows open to provide good air circulation. And please make sure you are wearing personal protection such as gloves, goggles and a paint mask. - EricL