While I am not an expert (and leaving aside seat/cabin considerations) either is very, very safe. BA are generally considered to have an extremely high level of safe operation.

The 747 appears to experience a few more technical issues that mean it does have quite the same level of reliability as the 777 but don't let that put you off.

Have you considered one of the 'fear of flying' courses offered by some airlines? A colleague did one a while ago and is now a comfortable 100,000 mile a year flyer.