Yes.
Let $M$ be any $A$-module. Then its socle is a direct sum of simple modules: $\operatorname{soc}A=\bigoplus_iS_i$.
$A$ is a finite dimensional algebra, so the dual $\mathrm{Hom}_k(A,k)$ of $A$ is a finite dimensional injective into which every simple module embeds. So each $I(S_i)$ is finite-dimensional.
The direct sum $I=\bigoplus_iI(S_i)$ of the injective envelopes of the simples is injective, so the natural inclusion $\operatorname{soc}A\to I$ extends to a map $M\to I$, which it is easy to see is an essential monomorphism, and so $I$ is the injective envelope of $M$.
If $M$ itself is injective, then $M=I=\bigoplus_iI(S_i)$ is a direct sum of finite dimensional injective modules.
