Alkali metals should have positive electron gain enthalpies as they are electropositive elements and also their atomic sizes are larger in their periods so they should be reluctant to take electrons but they show negative electron gain enthalpies. Why?
I think you mean to ask about the electron affinity of the alkali metals. That is, the enthalpy gain or lost when adding an electron to an atom.
While the electron affinity for alkali metals are much smaller than the halogens, they are slightly positive. That is, quoting from Wikipedia:
Electron capture for almost all non-noble gas atoms involves the release of energy and thus are exothermic.
The catch, is that a positive electron affinity (Eea) refers to an exo-thermic process:
Confusion arises in mistaking Eea for a change in energy, ΔE, in which case the positive values listed in tables would be for an endo- not exo-thermic process.
You refer to the alkali metals as "electropositive" -- in this case, you're referring to their electronegativity relative to, say, halogens. Yes, a sodium atom is likely to become a sodium cation and give up an electron. But the electron configuration is $3s^1$ and consequently, can also gain an electron and become the more stable $3s^2$ state.
It's more likely that a sodium atom will lose an electron, and become a cation, but the electron affinity reflects the modest gain in stability as $3s^2$ after attaching an extra electron.