If you really must rely on user-provided data that maps directly to path/filenames, and can't use a token system to represent the same thing, I would explicitely declare what your valid "root" directory is, and do a check like this:

use CGI ':standard';
use File::Spec 'rel2abs';
my $ROOT = "/var/myapp/docroot/";  # wherever

my $user_path = param('path');     # perhaps s/^\/+// also
my $absolute  = rel2abs($user_path, $ROOT);

if ($absolute =~ /^\Q$ROOT/) {
   # $absolute is probably within $ROOT, so process it
   if (open(INF, "< $absolute")) {
      # it's here, do whatever
   } else {
      # "404 not found"
   }
} else {
   # ERROR - They've tried to ../ their way out
}
[download]

Keep in mind, though, that this still lets them ../ their way anywhere they want under your declared $ROOT, so if you're expecting a filename to be in a certain place or under a certain hierarchy under your $ROOT, you need to do some additional checking/tokenizing to be sure that it actually does end up there. All this code does is keep the user sandboxed.

I too highly recommend reading perlsec and using taint-checking (-T) to better prepare yourself for potentially unsafe user-provided data.