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The answer to this question mainly comes down to which pair of these atoms has a larger electronegativity difference between them. Barium is down towards the left of the periodic table (where electronegativity is low) and Beryllium is higher up in the group than Barium. Thus, the difference in electronegativity between Ba and Br is larger than the difference between Be and Br, and therefore Br will effectively take electrons away from the Ba atom, giving its bonds a primarily ionic character.
In addition to the response above, the way we can tell the character of the bond has thresholds around 1.5, where less than that it will be covalent while greater it can be ionic character. There's a very nice diagram in that section that graphs out some specific examples and the character identification for compounds. The ionization energy calculation however is the most important thing.
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