One problem is that your hash is empty after you tried to enter the values - there is no assignment to the hash.

If you want to know the values of your hash, here's a little trick:

use Data::Dumper;
my %passwords; # please don't name your data structure
               # %new_hash. That's not a telling name at 
               # all
while(1)
{
        print "Please enter your first name\n";
        chomp ($fname =<STDIN>);
           {
                last;
           }
        print "Please enter your last name\n";
        chomp ($lname =<STDIN>);
           {
                last;
           }
        print "Please enter a password\n";
        chomp ($password =<STDIN>);
           {
                last;
           }
        print "Please enter 'done' or hit your return key\n";
        chomp ($ending =<STDIN>);
        if ($ending eq '\n' or 'done')
           {
                last;
           }
}

my $username = "$lname, $fname";

# here comes the magic:
print Dumper \%passwords; # you'll see that the hash 
                          # is empty.

# rest of the script goes here
[download]

You have to actually store the passwords:

while(1)
{
        print "Please enter your first name\n";
        chomp ($fname =<STDIN>);
           {
                last;
           }
        print "Please enter your last name\n";
        chomp ($lname =<STDIN>);
           {
                last;
           }
        print "Please enter a password\n";
        chomp ($password =<STDIN>);
           {
                last;
           }
        # important: store the passwords in %passwords
        $passwords{$fname} = $password;

        print "Please enter 'done' or hit your return key\n";
        chomp ($ending =<STDIN>);
        if ($ending eq '\n' or 'done')
           {
                last;
           }
}
[download]

Additionally the usual tips apply: always use strict; use warnings and declare your variables.

To answer your original question: to achieve case insensitivity, use lc or uc to convert the strings (for example first name) to all lower or all upper case, and only then store them.