Quote from matt_hooks:
...it would be better to lose the downwind engine, as the aircraft will have a tendency to turn towards the wind (weathercock) which you would need to counteract with rudder (I'm talking on a final approach of course, in level flight I don't think it makes any difference.)
[Unquote]
You are right that it's better to lose the downwind engine, but - generally speaking - there is no weathercocking when airborne. It is true, however, that it would help on final approach when using the side-slip crosswind-landing technique. Using the normal technique, it assists when decrabbing (kicking the drift off), immediately prior to touchdown.
In the engine failure on T/O (EFTO) case, it would help on the ground, particularly if the T/O was being continued, and during rotation. Once airborne, after allowing the aircraft to weathercock slightly to compensate for drift, it would make no difference.