Try this:

foreach my $hk ( keys %{ $href } ) { 
    print "$hk\n" for grep { /^$search$/ } @{ $href->{$hk}->{'member'}
+ };
}
[download]

#! perl
use strict;
use warnings;
use feature qw/ say /;

my $href = {
  foo => {
    member => [ 'fred', 'barney', 'wilma', 'betty' ],
  },
  bar => {
    member => [ 'mickey', 'minnie' ],
  },
};

my $search = 'wilma';

foreach my $hk ( keys %{ $href } ) {
    say $hk for grep { /^$search$/ } @{ $href->{$hk}->{'member'} };
}

__END__
[download]

Output:

[21:57][nick:~/monks]$ perl 1140148.pl
foo
[download]

Update: added output