One classic way is to just keep track of the most recent contender, and the one you're still holding at the end is the winner:

use File::Basename;

my $found;
while( <DATA> ) {
  chomp;
  my $bn = basename($_);
  next unless $bn =~ m/^\d+$/;
  $found = $bn;
}
print $found;

__DATA__
/path/to/one/library
/path/to/another
/path/to/another/01
/path/to/another/s
/path/to/another/02
/path/to/another/w1
/path/to/another/03
/path/to/another/1e
/path/to/another/2014
[download]

If the file is large enough that you don't want to spend time reading the whole thing, you could use a File::ReadBackwards solution:

# Untested...
use File::Basename;
use File::ReadBackwards;

tie *FH, 'ReadBackwards', 'filename' or die $!;

my $found;
while( <FH> ) {
  chomp;
  next unless basename($_) =~ m/^(\d+)$/;
  $found = $1;
  last;
}

die "No matches found.\n" unless defined $found;
print $found;
[download]

Or if the file's not huge, you could just slurp it into an array and reverse the array:

use File::Basename;

my @files = reverse <DATA>;
my $found;
foreach ( @files ) {
  chomp;
  next unless basename($_) =~ m/^(\d+)$/;
  $found = $1;
  last;
}

die "No matches found.\n" unless defined $found;
print $found;

__DATA__
/path/to/one/library
/path/to/another
/path/to/another/01
/path/to/another/s
/path/to/another/02
/path/to/another/w1
/path/to/another/03
/path/to/another/1e
/path/to/another/2014
[download]

Of course if you're going to slurp it, you probably don't really need the @files variable:

foreach ( reverse <DATA> ) {
  ...
[download]

This still slurps, and people generally advise against iterating over a filehandle within a foreach loop, because the foreach loop does implicitly slurp the file. But in this case, that's the behavior we actually want.