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:
R87 983 per person with Lufthansa
R168 009 per person with SWISS
R171 440 per person with KLM