To download the vSphere Client for ESXi 5.3 or 5.5, you'll need to visit VMware's official site or search for the exact version in their archives, as those older clients aren't readily available anymore.
VMware vSphere 5.1 is now released and available for download. Customers who have purchased VMware vSphere 5.1 can download their relevant installation package from the product download tab below.