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I want to prove that $^{38}Ar$ is stable with respect to $\beta$ decay, that $^{38}Cl$ decays by $\beta^-$ and that $^{38}K$ decays by $\beta^+$.

I know from Googling that this is true, and I also know that $^{38}Ar$ has the highest binding energy for mass number 38, thus its definitely stable with respect to $\beta$ decay.

However I would like to prove this from a method other than calculating out all the binding energies from the semi empirical mass formula, is there and easier way?

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I think your question comes down to asking for a complete theory of nuclear structure. I'm not aware of a solution for you (aside from using the empirical masses, but I assume that's not what you want either). –  dmckee Feb 21 '13 at 0:27
The fact that a semi empirical formula is in use shows that a complete theory does not exist. –  anna v Feb 21 '13 at 6:02

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