Alkali metals should have positive electron gain enthalpy as they are electropositive elements and also there atomic size is big 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.
After taking one electron ns2 orbital is fulfilled and become stable. So Alkali metals has negative electron gain enthalpy.
All alkali metals have negative electron gain enthalpy. Ionisation of sodium(or any atom) is endothermic process (495.8 KJ/mol), meaning, in isolated gaseous state, neutral atom is more stable, Electron gain process of Na+ is obviously an exothermic process (reverse of ionisation) due to attraction of Positively charged, Na+ for a free electron. If an atom has spontaneous tendency, i.e., a positive tendency , to gain electron, then the electron-gain process is exothermic & hence, its electron gain enthalpy is said to be negative and if the atom is reluctant to gain an electron, i.e., it has a negative tendency to gain an electron and is forced to accept it, the electron-gain process is exothermic & hence its electron gain enthalpy value is positive. Electron gain enthalphy of Na+ is (-495.8) KJ/mol. [process is Na+(g) + e= Na(g)] Note that here, obviously process is related to gaseous phase. So, any misconception about , like non-spontaneous electrolysis of NaCl (is due to compensation of ionisation energy of sodium by greater stabilisation of ion by hydration) or, Stability of sodium compoounds in which sodium ion is present (Ionisation energy is compensated by lattice energy of the solid) is common.