Metal sinks with porcelain coating baked on at high temperatures tend to rust over time and the coatings often end up chipped or scratched. Left as is, the sink will continue to deteriorate, rust and possibly leak.
Finger nail polish
Touch up kits. There are a number of touch-up and repair kits on the market to help touch-up these nicks, chips and scratches in these sinks. They work a little like putting on finger nail polish and come in a number of colors, so you can match the sink surface.
First check the fittings, drain and water supply connections. If the connections are ok, then check to see if an area of the sink has rusted through or if on the under-side there may be an overflow channel on the bottom of the sink leaking. At times the leaking over flow channel can be silicone caulked or sealed to prevent the dripping or leaking.
Rusted sinks may be repaired at times, however, if the rust is severe or has entirely rusted through the metal, then replacing the sink is a good idea. These types of sinks are usually very inexpensive.
Who to consult? Handymen and plumbers.
Home owners can do much of the maintenance on a sink and even replace it if they wish. Handymen and plumbers can generally do the work quickly and in-expensively.