There are no safety concerns in Mar del Plata that are any different from those present in other parts of Argentina. It is a big city and, especially during the summer crowds, can be very crowded in the main public areas. Be aware of pickpockets.
Even at night Mar del Plata is a relatively safe city but it is recommended to take a taxi back from the nightclubs on Constitution or anywhere else not centrally located. The city is also one of the only in Argentina the permits beach access at night. This is a great aspect of the summer life here, but woman and anyone alone should be cautious of less populated areas of the beach late at night.