In anything other than clear daylight or electronic flash, all daylight-balanced color film - positive or negative - has a color cast. It is balanced to be neutral under these light sources alone. The more saturated and contrasty a film is, the more severe this cast appears. As such, Velvia exaggerates these casts.
If you shot your film in clear sunlight or with electronic flash, then I would look at the processing, or age and/or storage conditions.
At any rate, the exposure does not significantly affect the color cast when compared to the other factors that do so, so I would rule that out.
When shooting negatives, you don't often see the cast unless you are doing your own printing (meaning that you know your neutral daylight filter pack for the film you are using), or examining your negs with a color densitometer. This is because you are looking at prints, which are always (and very easily) altered to get them into positive form. With a transparency, you get the same casts as you get with negatives, but you are looking at a positive image right on the film, without any printing adjustments.
Going back to my first statement about the color of light in which you shoot affecting your picture, FYI, the same applies to black and white film. The difference is that you do no see the results as a color cast (obviously), but as an apparent difference in spectral response, similar to using colored filters in front of your camera lens. For instance, shoot a b/w shot in open shade, and your film will be more exposed by blue and green light than it will be by red light. Therefore, it will respond to color similarly to the way it would in the Sun with a cool color balancing filter (such as a #82A) in front of the lens.