Most common way is to use basic auth - this is how browsers 'store' passwords. The server asks for authentication, the browser sends it, and keeps sending it in subsequent requests. Usually, this doesn't get saved anywhere unless you ask for it. Of course, the pages may be subject to cacheing though.

The encryption / non encryption thing is mostly about preventing people sniffing your traffic - not highly likely, unless you have reason to suspect otherwise - but the main worry is probably leaving stuff behind in cache. Basic auth is probably fine for most uses, esp. over SSL, but I suppose it depends how paranoid you are...