Background: The synthesis of heavy, proton rich isotopes in the astrophysical γ process proceeds through photodisintegration reactions. For the improved understanding of the process, the rates of the involved nuclear reactions must be known. The reaction Ba128(γ,α)Xe124 was found to affect the abundance of the p nucleus Xe124 in previous rate variation studies. Purpose: Since the stellar rate for this reaction cannot be determined by a measurement directly, the aim of the present work was to measure the cross section of the inverse Xe124(α,γ)Ba128 reaction and to compare the results with statistical model predictions used in astrophysical networks. Modified nuclear input can then be used to provide an improved stellar reaction rate. Of great importance is the fact that data below the (α,n) threshold was obtained. Studying simultaneously the Xe124(α,n)Ba127 reaction channel at higher energy allowed to further identify the source of a discrepancy between data and prediction. Method: The Xe124(α,γ)Ba128 and Xe124(α,n)Ba127 cross sections were measured with the activation method using a thin window Xe124 gas cell and an α beam from a cyclotron accelerator. The studied energy range was between Eα=11 and 15 MeV close above the astrophysically relevant energy range. Results: The obtained cross sections are compared with Hauser-Feshbach statistical model calculations. The experimental cross sections are smaller than standard predictions previously used in astrophysical calculations. As a dominating source of the difference, the theoretical α width was identified. The experimental data suggest an α width lower by at least a factor of 0.125 in the astrophysically important energy range. Conclusions: An upper limit for the Ba128(γ,α)Xe124 stellar rate was inferred from our measurement. The impact of this rate and lower rates was studied in two different models for core-collapse supernova explosions of 25 M stars. A significant contribution to the Xe124 abundance via this reaction path would only be possible when the rate was increased above the previous standard value. Since the experimental data rule this out, they also demonstrate the closure of this production path.