If ~ really is hardcoded in the script, you'll need to expand it first. perlfaq5 has a nice regex for doing just that:

$filename = "~/madam";
$filename =~ s{
    ^ ~     # find a leading tilde
    (       # save this in $1
    [^/]    # a non-slash character
    *       # repeated 0 or more times (0 means me)
    )
  }{
    $1
    ? (getpwnam($1))[7]
    : ( $ENV{HOME} || $ENV{LOGDIR} )
  }ex;

print $filename, "\n";
[download]

Update and to combine the regex with your code:

#!/usr/bin/perl

use strict;
use File::Spec;

my $str = " /home/madam  ~madam ~/madam";
my @strarray = split(" ",$str);

for (@strarray){
  my $file = expand_tilde( $_ );
  if ( -d $file ){
    print "path present : $file\n";
  } else {
    print "path not present : $file\n";
  }
}

sub expand_tilde {
  my( $file ) = shift;
  $file =~ s{
    ^ ~             # find a leading tilde
    (               # save this in $1
    [^/]        # a non-slash character
    *     # repeated 0 or more times (0 means me)
    )
  }{
    $1
    ? (getpwnam($1))[7]
    : ( $ENV{HOME} || $ENV{LOGDIR} )
  }ex;

  $file
}
[download]