Make sure that water is not pooling in that plastic-line basket. Water the plant in the sink and let it drain there, or else tip out that basket immediately after watering. Any saucer (or that basket) should be used only to catch drips and condensation.
You certainly can re-pot your Corn-plant (Dracaena fragrans), but you many not have room on your window sill. Also, the plant is already too tall for the window. If you cannot set the plant on the floor or other lower surface, jut persist as is, but do check on the plant regularly. You will want to maintain slightly damp soil conditions with good drainage and air to the roots
Dracaena fragrans, in the Butcher's-broom Family (Ruscaceae; alternatively Nolinoideae of Asparagaceae), is native to tropical Africa.