Why basicity of oxides increases down a group?
As suggested in comments, major factors play a part


Atomic size


As the atomic size of the element increases, its electron affinity decreases consequently. It turns out that when moving vertically in the periodic table, the size of the atom outdoes its electronegativity with regard to basicity. The atomic radius of gallium, for example is approximately twice that of boron, so in a gallium ion, the charge is spread out over a significantly larger volume.

Electrostatic charges, whether positive or negative, are more stable when they are ‘spread out’ than when they are confined to one location.


Electron affinity


The ability of an atom “to accept an electron” decreases down the group therefore increases the tendency of the elements to be good Lewis bases (recall a Lewis base is an electron-pair donor). Thus the decrease in electron affinity down the group means the elements readily donate an electron rather than accept them.


Electronegativity 


The more electronegative an atom, the better it is able to bear a negative charge. Down the group, the electronegativity of the elements decrease down the group.  B is very electronegative and doesn’t readily release electrons but instead holds on to it, additionally is able to stabilise a negative charge, and consequently has good acidic properties unlike proceeding elements down the group.