The closest airport to Mamaia Beach is Constanţa Airport, which is located in Constanţa. If you plan on visiting or staying in this area, landing at Constanţa Airport will be your best bet.
The best prices found on momondo for flights to Romania are:
R1 870 per person with Turkish Airlines
R4 560 per person with Lufthansa
R8 888 per person with KLM