I don't think you meant to use the same letter for the name of the space and the name of the index set, did you? I'll use E for the space and A for the index set.

If A is finite then E is finite-dimensional. Suppose, for a contradiction, that E is infinite-dimensional but has a countable base {e_n} (indexed by the natural numbers). For n=1, 2, ..., let E_n be the subspace of E spanned by the first n basis vectors e_1, ..., e_n. Let A_n be the complement of E_n in E. Then A_n is open and dense (easily proved). So by Baire's theorem the intersection of the A_n is dense. But every element of E is in E_n for n large enough, so the intersection of the A_n is empty. Contradiction.