Order of basicity of halides
The electron gain enthalpy you've mentioned is measured in the gas phase, not in aqueous solution like the order of basicity.

When you go down on a group, two important properties are increased: nuclear charge and atomic radius. However, the increase in atomic radius is much more pronounced than of the nuclear charge. Considering this, as you go down on the halogens group the electrons added feel less attracted to the nucleus and, therefore, the electron gain enthalpy become less negative.

The order is only apparently changed for $\ce{Cl-}$ and $\ce{F-}$ due to fluorine small size. The additional electron would be added to a "small" $\ce{2p}$ orbital in fluorine (and there would be more electron-electron repulsion) while in the chlorine atom it would be added to a "big" $\ce{3p}$ orbital that can better accommodate the additional charge, so the enthalpy is less negative for the electron gain in $\ce{F}$ then for $\ce{Cl}$.