Use lc to make your input case-insensitive:

use warnings;
use strict;

my @password = ('foo');

print "Do you want to export your password to a text file? (y/n) ";
chomp(my $export = <STDIN>);
$export = lc $export;

if ($export eq 'y') {
    open (FH, '>>password.txt') or die $!;
    print FH "Your password is: ", @password, "\n";
    close FH;
} elsif ($export eq 'n') {
    print "Your password is: ", @password, "\n";
} else {
    print "Enter either y or n to continue. \n" ;
}
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You do use strict and warnings, don't you? See also Equality Operators.