Fix The Leak FIRST
Before you get started on removing mold and mildew under a sink, you'll need to first find and fix the leak causing the moisture in the cabinet. Under a kitchen sink the source could be anything from a leak in a water or drain pipe to a leaking garbage disposal or dishwasher. It might even be caused by water seeping under the kitchen sink and dripping into the cabinet below.
Mold will only grow where there is moisture, which is why it's so important to take care of the source of the moisture. Moisture can also come from items such as wet or damp rags, so it's important to allow them to dry out before storing them in a cabinet. Once you've taken care of the source of the moisture, you can focus on removing mold and mildew under the sink.