Cron, generally, sends email to its owner when it runs anyway, so the first thing you should do is make sure that email to the "owner" is directed to a mailbox you have access to. Depending on your hosting situation, that could be easy, or it could be nigh impossible.
As NogDog suggests, debugging it will be harder than reading normal job output. However, if you can code your jobs to handle most types of errors with tests, exceptions, trigger_error(), what-not, it's possible you could also get that in your email.
Here's an email I received recently from a cronjob, and I didn't really have to code anything:
PHP Fatal error: Uncaught exception 'GrabzItException' with message 'The screenshot did not complete with the error: ' in
#0 /var/www/apache24/data/my.com/html/GrabzIt/fetch_screenies.php(48): GrabzItClient->SaveTo('/usr/local/www/...')
thrown in /var/www/apache24/data/my.com/html/GrabzIt/php/lib/GrabzItClient.class.php on line 219
Unfortunately it's not OVERLY informative. But it's a start.