Browser add-ons are browser-specific, so any answer to your question can only be very general.
The biggest problem with a browser add-on is that not everybody has it. If people must install an add-on to use your Web site, then this no longer is a Web site; the whole point of a Web site is that everybody can use it without any prior installation. An add-on is useful only when it provides optional client-side functionality (so your site code cannot rely on its presence), or when everybody already has it (in which case it no longer is an add-on, but a "native feature").
The two "add-ons" which got close to being sufficiently prevalent to be considered as "always there" are Flash and Java, but both are still restrictive (an iPad, for instance, has neither). It would be quite improbable for your add-on to achieve an even similar level of success. This means that whatever reasons which would make doing your browser-side cryptography in a Java applet a bad idea, also apply to the scenario of using a crypto add-on (and Java, at least, has the muscle and existing code base which allows for powerful cryptography).