What do you get when you add:

print "@ids\n";
print "@files\n";
[download]

to your code?

This doesn't seem like the right approach to me:

foreach my $id (@ids) {
    foreach my $fasta (@files) {
        open my $fh, '<', $fasta or die "can not open file $fasta: $!"
+;
        while (<$fh>) {
            print if /$id/;
        }
        close $fh;
    }
}
[download]

If there are 5 ids and 10 files, that code will open and close each file 5 times. How about something like this:

my $pattern = join '|', @ids;
print grep {/$pattern/} <>;
[download]