I've found the best way to make upgrading easier is having good test coverage. I think one of the biggest fears when upgrading is having something break but not knowing that it is now broken. Although tests won't always catch everything they are a great start to ease the upgrade process.
To have tests (and particularly integration tests) is a big help; as has already been mentioned.
Another thing is to use staged deployments: dont deploy to production-env from you development-env, but to a staging-env (if possible using the same or a copy of the live database). Then have a look with some (possibly non-technical) humans if the app still behaves as expected.
While it can be laborious for a large app, it should make a deployment to production a lot less scary.
As mentioned by others: upgrading is part of running a web app that depends on popular libs.
Why does it scare you? It's good to be wary of the upgrade process.. but this is why you have (I hope!) database backups and source control (as a minimum). Then at least you can roll back. Then it's just extras after that, test coverage, staging servers etc. etc.
Anyway, upgrading is a fact of life once you're using a framework (unless you want to get hacked) so best to get used to it :)