dmartin - Unless there is a strong draft, pilots don't normally don't blow out.
What you described, sounds more like the interior of the heater needs a thorough cleaning. Especially, the pilot assembly and the Oxygen Depletion Sensor (ODS), which is part of the pilot. Also, the placement of the gas grill tank you're using to fuel the heater, may also being playing a role in it, too.
How far away is the tank from the heater and what size gas line is running from the tank to the heater and was it professionally installed? If you live in a high altitude are (above 2000 feet) this may also, be causing the problem.
I have included a link, which demonstrates the proper way to clean a pilot assembly and the ODS sensor. This should be a big help to you. http://www.desatech.com/techDocs/Cleaning%20Demonstrations.pdf
Please let me know about my questions above and I can provide you with more information on further troubleshooting the problem.