Step 1: The basic strength of oxides generally increases as we move down a group in the periodic table. This is because the oxides become more ionic and thus more basic.
Step 2: The order of basicity is inversely related to the electronegativity of the metal. As we move from Be to Ba, the metal's electronegativity decreases, and the oxide becomes more basic.
Step 3: The correct order of increasing basic strength is: \[ {BeO} < {MgO} < {CaO} < {BaO}. \]