Não consigo executar nenhum applet Java em todos os navegadores.
Firefox:
O Firefox 64 bits não suporta o JRE.
O Firefox 32 ainda suporta o JRE (mas o suporte será retirado em 2016). Você precisa instalar o JRE de 32 bits para que isso funcione.
Mozilla intends to remove support for most NPAPI plugins in Firefox by the end of 2016. Firefox began this process several years ago with manual plugin activation, allowing users to activate plugins only when they were necessary. This decision mirrors actions by other modern browsers, such as Google Chrome and Microsoft Edge, which have already removed support for legacy plugins.
Moreover, since new Firefox platforms do not have to support an existing ecosystem of users and plugins, new platforms such as 64-bit Firefox for Windows will launch without plugin support.
Fonte Plugins NPAPI no Firefox
Chrome
O Chrome não suporta mais o JRE.
Java is a NPAPI plugin.
NPAPI plugins are being phased out of Chrome 42. Please note that Chrome will not support NPAPI plugins sometime after April 2015, therefore any websites that use these plugins will all stop working: It will get more difficult to run NPAPI plugins in Chrome.
Fonte Os plug-ins Java desaparecem após a nova atualização do Chrome para a versão 42.0.2311.90 m