We have flown numerous times with and without babies, twins, toddlers and even grandparents with BA, BMI, Virgin and United from Heathrow/Manchester to Washington/New York ....
Virign WINS every time - the most helpful staff at check-in and on board - they actually seem to enjoy their jobs - great in flight entertainment, good food etc, etc ... they make the best of what's a pretty awful experience of being stuck in a metal tube for 8 hrs, hurtling along at 500 mph - UGH I HATE FLYING!!!!
As for the others, well, they've all been pretty good too.
BA has a better schedule than Virgin, the service is great when it's great but we have come across the odd BA snotball (I think they had big union problems a couple of years ago?) - they lost our luggage once, but they have upgraded us a number of times (No idea why - we usually look like a pair of deadbeats).
United - ditto above but they used to have a great frequent flyer program - we had a lot of free trips to Paris on the Eurostar.
BMI are 'quaint' - nice staff - they only recently started flying Transatlantic and still seem excited about it!!
Oh, I think what can affect a flight is the airport experience - I have to say Heathrow is just about the worst airport ever - it's SOOOOOOOOOO overcrowded and stuffy - Manchester is 100 times better and puts me in a calmer mood before flying.