I recently stumbled onto Cheri Down Park at 8400 Ridgewood Ave in Cape Canaveral. I had never heard of it before and it turned out to be a really-nice, quiet, serene place with white sand - and a great view of both Port Canaveral's traffic - and of Cape Canaveral. I label it a "find".
It is easy-access: just take the "Beach Line (528)" straight over from Orlando - right into A1A - until just-before it turns South. Refer to the map - taken, as usual, from Microsoft's "Streets & Trips". 
Cheri Down is a few miles north of famous Cocoa Beach - so you can actually see Cocoa Beach pier from Cheri Down - but you don't have as much of an immersion in "the tourist experience": it feels more quiet and special. It is also just South of the entrance to Port Canaveral, so we got an interesting, slow, progression of cruise ships, tankers, and even a *submarine* churning out to sea - during the day I was there. There is plenty of parking and even restrooms available - before you head over the boardwalk to the sandy beach.
The gantries of the Space Center are also clearly visible, so this would be a great vantage point for watching shuttle or rocket launches - like the Shuttle Discovery - currently scheduled to launch on Feb 27. Note in the picture that you can see the Space Center gantries behind the submarine.

Just pack a lunch, head over the Beach Line - and stop at a lesser-used Beach for a quieter day.