Hello.
If you choose the VMware client (HTML5) you must use a browser that supports HTML5, most of them in their new versions support it.
If you choose the VMware Web Client (Flash), you must use a 32bits (old) browser with Flash support.
I work without problems in versions 6.0 and 6.5 with the following configuration:
Windows 8.1
Firefox 38.8 (32bits)
Adobe Flash 29.0.0.113
VMware Client Integration Plugin 6.X included in the VCSA ISO
Firefox 38.8 I only use it when I have to manage in VMware, for other things I use a more modern and updated browser
Attached link about web client and HTML5 available from version 6.5 to version 6.7
vSphere Client (HTML5) and vSphere Web Client 6.5 FAQ (2147929)
https://kb.vmware.com/s/article/2147929?lang=en_us