Yes it will. It is additional computing effort and naturally it will use more resources. But it's only a very tiny fraction of the total request processing time.
You won't lose the benefits of caching. Server side caching is all about how you have designed your application. Browsers cache content even if they are delivered through https.
I haven't developed 'high-traffic' applications in their general sense. But in the applications that I have developed so far, I could not observe any difference at all in performance or user experience.
Also, if my application does suffer any performance loss because of SSL, security would still be my priority over performance.