{
  package MyAgent;

  use vars qw(@ISA);
  @ISA = qw(LWP::UserAgent);

  sub new {
    my $self = LWP::UserAgent::new(@_);
    $self->agent("lwp-request/$LWP::UserAgent::VERSION");
    $self;
  }

  # this routine gets called when your browser
  #   would otherwise be asked to provide a
  #   username and password
  sub get_basic_credentials {
    my ($self, $realm, $uri) = @_;
    print( STDERR "  - providing auth to realm \"$realm\"\n" );
    return( $username, $password );
  }
}
[download]

{
  package MyMech;

  use vars qw(@ISA);
  @ISA = qw(WWW::Mechanize);

  sub new {
    my $self = WWW::Mechanize::new(@_);
    $self->agent("www-mechanize/$WWW::Mechanize::VERSION");
    $self;
  }

  # this routine gets called when your browser
  #   would otherwise be asked to provide a
  #   username and password
  sub get_basic_credentials {
    my ($self, $realm, $uri) = @_;
    print( STDERR "  - providing auth to realm \"$realm\"\n" );
    return( $username, $password );
  }
}
[download]

You just then use your new class instead.

  my $mech = MyMech->new();
  $mech->get( $url );
  print $mech->status, "\n";
  print $mech->success;
[download]

Note this technique not only works with web sites using Basic but company web proxies that stop and ask you for a username and password before proceeding (it appears this could be the case for this particular thread).

Update:
- Added readmore tags..
- Added comment about company proxies