I raised this same question last week. Must admit the documentation seemed vague - I ended up looking though the pm file! Anyway I got it working with something like..

use LWP::UserAgent;
#..... usual stuff .....#
$ua = new LWP::UserAgent;
$ua->proxy(['http', 'ftp'] => 'http://your_proxy.address.com:');

my $req = new HTTP::Request 'GET',
    "http://www.perlmonks.org";

$req->proxy_authorization_basic("$user_name","$user_password");

$res = $ua->request($req);
[download]

Update - I've just seen that chromatic posted a virtually identical code snipet in answer to my original question.
Never saw it at the time Doh! That will teach me to give up on the perl monks.