Moderator: Please try and create a thread in the area for the product used - moved to vSphere Discussions
The comments and links in this thread may help: https://communities.vmware.com/t5/VMware-vSphere-Discussions/vSphere-Flash-EOL/m-p/2822292#M37619
Isn't vsphere client works with vcenter 6.0 ? if you use 6.0
Also, html5 client is also available in 6.5
If you still want to use flash client, then follow instructions by scott.
In environments with old versions there is no other option than to use a browser that supports Adobe Flash.
Personally I use the following combination, but only to manage VMware, to surf the internet I use a more current browser.
Firefox ESR 38.8 (32bits)
Adobe Flash 188.8.131.52
VMware Client Integration Plugin 6.0 included in the VCSA ISO
To login using the vSphere Web Client (Flash). You can instead open a Web browser and enter the URL for the vSphere Client: https:// ip_address/ui (A browser with Adobe Flash and VMware CIP is required)
To use the vSphere Web Client (HTML5) , enter the URL: https://ip_address/vsphere-client . (A browser HTML5 is required)