Emacs 24.3 or below

There's no built-in way of preventing these old files from being loaded, but there are easy ways to get rid of them.

• You can recompile the entire elpa directory by calling:
  M-x byte-recompile-directory RET ~/.emacs.d/elpa/.
  This should get rid of outdated files.
• You can use the auto-compile package and activate auto-compile-on-load-mode which can compile files before they are loaded.

Emacs 24.4

Yes, and it turns out to be rather simple. The load-prefer-newer variable serves precisely this purpose.

(setq load-prefer-newer t)

Unfortunately, it won't work when some code specifically targets the .elc file, such as (load "server.elc"). But it should be enough as long as you're using requires and notor calling load without a suffix, which you should.

From the doc:

load-prefer-newer is a variable defined in lread.c.
Its value is nil

Documentation:
Non-nil means load prefers the newest version of a file.
This applies when a filename suffix is not explicitly specified and load is trying various possible suffixes (see load-suffixes and load-file-rep-suffixes). Normally, it stops at the first file that exists unless you explicitly specify one or the other. If this option is non-nil, it checks all suffixes and uses whichever file is newest.
Note that if you customize this, obviously it will not affect files that are loaded before your customizations are read!