Statement-1: Electron gain enthalpy generally becomes less negative as we go down a group. Statement-2: Size of the atom increases on going down the group and the added electron would be further from the nucleus.

A claim cannot be true.
As we move down a group in the modern periodic table, the electron gain enthalpy does not always get less negative.
The Modern Periodic Table's explanation, according to which the size of the atom rises as you move down the groups and an additional electron moves away from the nucleus, is true.