Step 1: Know the group 2 trends.
Going down group 2 (Be to Ba), the metal ions get bigger and more electropositive, so their hydroxides become more soluble and more basic.
Step 2: Check the basic character option.
For the hydroxides, basic strength rises down the group, so $Mg(OH)_2 < Ca(OH)_2 < Ba(OH)_2$ is the correct increasing order.
Step 3: Test the covalent character option.
Covalent character falls down the group (Fajans rules), so the correct order would be $CaCl_2 < MgCl_2 < BeCl_2$. The given $BeCl_2 < MgCl_2 < CaCl_2$ is reversed, so it is wrong.
Step 4: Test the solubility option.
Sulphate solubility decreases down the group, so $BaSO_4 < SrSO_4 < CaSO_4$. The given order is backwards, so it is wrong.
Step 5: Test the thermal stability option.
Carbonate thermal stability increases down the group, so $MgCO_3 < SrCO_3 < BaCO_3$. The given order is reversed, so it is wrong.
Step 6: Conclusion.
Only the basic character order is correct: $Mg(OH)_2 < Ca(OH)_2 < Ba(OH)_2$. \[ \boxed{Mg(OH)_2 < Ca(OH)_2 < Ba(OH)_2} \]