This actually depends on quite a few factors. I wondered this once many years ago, and asked around quite a bit. Didn't have Travel.SE back then ;)
The earth is rotating at a rather fast speed - and any point on the earth is therefore actually 'moving' (it's all relative). Since the points on the equator have further to travel, they're moving even faster than at the poles.
Now, of course, the air is dragged around WITH the earth, thankfully, otherwise the poor chaps on the equator would have wind speeds in the opposite direction of near the speed of sound ;)
However, when you're in a plane, consider that it can take nearly an hour longer to fly across the Atlantic in a westerly direction ('against' the spin) than 'with' the spin.
When you're flying with the spin, and by relation, with the wind, you're not flying 'into' a force that's going the other way, as you are when you fly against the spin. The earth is also dragging you with it - or rather, it's dragging the atmosphere, and you in it.
However, what you'll tend to find is that it's actually far more dependent in reality on the existence of jetstreams - where the air up there is moving faster than at ground level, and can boost the plane's speed if going in the same direction. Of course, in the other direction you do well to avoid the jetstream, as it would slow you down.
To put it in words more eloquent than my own, I'll borrow a quote from Aerospaceweb.org, which first, you must consider yourself to be running....
Stop running. If you were to jump straight up in the air, would the
Earth rotate beneath you? (Those who do believe that the Earth rotates
around them may want to stop reading right now.) No, because when you
left the Earth's surface, you were traveling at the same speed as the
surface, so, in essence, the Earth matched your speed through space
while you were in the air! The same condition holds true for an
airplane as it travels from Los Angeles to Bombay. If we were to
ignore the winds, no matter which direction you flew from Los Angeles,
the speed of the aircraft relative to the Earth would be the same.
While the aircraft's speed through space would change, the effect of
the Earth's rotation remains constant, and in effect is "cancelled
out" no matter which direction you travel. In other words, the speed
of the rotation of the Earth is already imparted to the aircraft, and
the Earth matches that speed during the entire flight. (Of course, in
the case of spacecraft, these speeds become very important.)
So, the end result of that long discussion is that the rotation of the
Earth has no effect on the travel time of an aircraft. Actually, it is the headwinds and tailwinds that cause the change
in travel times. Sometimes it is hard to believe that the winds can
have that much effect, so let us consider the problem a bit more in
depth. In the example given, the flight from Bombay to California
(east) is 23% shorter than the trip from California to Bombay (west).
This means that the speed of the trip east must be 23% faster. The
prevailing winds pretty much anywhere that we are talking about blow
from west to east, so when we are traveling east, we get a speed gain,
and when we travel west, we get a speed penalty. Now, if we are to
assume that the winds are identical on both days we fly, then the wind
speed only needs to be equal to 11.5% of the aircraft's speed! This
would cause a difference between the speed to the west and speed to
the east of 23%! The cruise speed of the extended range Boeing 777 is
about 550 mph (885 km/h) at 35,000 ft (10,675 m). This means that the
winds only need a speed of about 65 mph (105 km/h) (good kite
weather). Believe it or not, 65 mph is a very typical wind speed at
such a high altitude. Speeds of over 100 mph (160 km/h) are not
uncommon. If we wanted to make things more complicated, we could
consider a region of high speed flow called the jet stream that flows
eastward, and if an aircraft can take advantage of these winds, then
the travel time can be reduced further.
Also note this LIVE amazing display of the prevailing winds in the USA, which affect all this.
So what is the bottom line? The direction you travel in relation to the rotation of the Earth does not affect the travel time of an aircraft, and, more importantly, a mere 65 mph wind is more than enough to cause a difference in travel time of five hours when you are traveling long distances!