A (closed) subspace $ Y$ of a Banach space $ X$ is called proximinal if for every $ x\in X$ there exists some $ y\in Y$ such that $ \|x-y\|\le\|x-z\|$ for $ z\in Y$. It is the object of this paper is to study the proximinality of $ L^\Phi(I,Y)$ in $ L^\Phi(I,X)$ for some class of Young's functions $ \Phi$, where $ I$ is the unit interval. We prove (among other results) that if $ Y$ is a separable proximinal subspace of $ X$, then $ L^\Phi(I,Y)$ is proximinal in $ L^\Phi(I,X)$.