Plurality will not be changed by using "the" since the subject is still "flights".
As to the article, it will actually change the meaning:
Over half of the domestic flights were delayed because of bad weather conditions.
The subject is "flights", and "the" specifies which (subset of) flights we are talking about. "Domestic" also narrows it down further, but without "the" we change the meaning:
Over half of domestic flights were delayed because of bad weather conditions.
Without a definite article, this means (half of) all domestic flights anywhere. Unless you mean that world-wide weather conditions are at play, or you specify "from this airport" you probably don't mean what this sentence says.
Using an article tells us that you are only referring to a specific subset of objects. We may need to infer what that subset is from context. If we are standing in an airport, if we are talking about a specific flight previously, if we are discussing or witnessing the weather in a certain place - we can infer the meaning. Without the article we can only tell that you didn't mean "all flights anywhere" because it's absurd.