The closest beach to Port Canaveral Pier is Cocoa Beach. Or so I've heard! Other than Port Canaveral beaches that is. Sailing on the July 3 MIracle and we definitely want to stay close to the ship! They offer transportation/packages to the Orlando theme parks, with our luck we would get stranded in Disney and miss the ship!!!
Alright!! A subject I finally feel I am an expert on!!! I live in Orlando and regularly go to beaches over by Pt. Canaveral (this weekend upcoming, in fact!). Now if you expect clear Caribbean blue waters, you won't get it this far north, but there are decent beaches to go to, depending on what you are looking for. Right by the Port is Jetty Park. It's about a mile or two from the pier. The park is clean and good for families. They do have a little snack shop, bait shop, etc. It is a campground, but you can go and use the beach (you don't have to camp out). You can also go walk out on to the end of the jetty and see the boats coming in and out of the port.
If you want more of a "party" atmosphere (although if it's not the weekend, I don't know how "partylike" it will be), I suggest going to the beach at the Cocoa Beach Pier or the public beach in Downtown Cocoa Beach right out front this waterfront restaurant/bar called Coconuts. On the weekend there is live music at Coconuts. You can also get tasty frozen drinks and take them with you on the beach. The same goes for the Cocoa Beach Pier (except it has a pier over water where the restaurant/bars are). While there are usually more people at these locations, I find these two beaches aren't as clean as Jetty Park.
I believe Jetty Park is $5 per car, per day. The other beaches have nominal parking fees as well. Please let me know if you have any questions.
New Smyrna is about 40 minutes north of the port and Daytona is just over an hour.