According to the information I can find, the net contribution of the Black Sea to the Mediterranean sea is about 5000 cubic meter per second
The Black Sea experiences water transfer only with the Mediterranean Sea, so all inflow and outflow occurs in the Bosporus and Dardanelles. Inflow from the Mediterranean has a higher salinity and density than the outflow, creating the classic estuarine circulation. This means that the inflow of dense water from the Mediterranean occurs at the bottom of the basin while the outflow of fresher Black Sea surface-water into the Marmara Sea occurs near the surface. The outflow is 16,000 m3/s (570,000 cu ft/s) (around 500 km3/a or 120 cubic miles per year) and the inflow is 11,000 m3/s (390,000 cu ft/s) (around 350 km3/a or 84 cubic miles per year), according to Gregg (2002).
On the other hand, the net flow through Gibraltar strait amount to 0.038 Sv, with 1 Sv being 1 million cubic meter per second, thus 38000 m3/s.
The net flow through the strait, estimated from the Mediterranean water budget, and the Mediterranean outflow, estimated from currentmeter observations in Espartel sill (western Strait of Gibraltar) from October 2004 to January 2009, made it possible to estimate the Atlantic inflow as the sum of both of them. The obtained mean net flow is 0.038 ± 0.007 Sv, with a seasonal cycle of 0.042 ± 0.018 Sv annual amplitude and maximum in September. The Mediterranean outflow shows a seasonal signal with annual amplitude of 0.027 ± 0.015 Sv peaking in April (in absolute value), and a mean value of −0.78 ± 0.05 Sv. The resulting Atlantic inflow has a mean value of 0.81 ± 0.06 Sv and a seasonal cycle with annual amplitude of 0.034 ± 0.011 Sv, peaking in September, and high interannual variability.
I don't think that damming the Black sea alone will make a dent in the Mediterranean sea level: the same paper I cited above mentions that the average Atlantic inflow is 0.81 Sv, and that means that the flow through Gibraltar can easily accommodate the missing inflow from the Black Sea, simply by reducing or nullifying the Mediterranean outflow.