Drowning has little to do with it, as crossing running water would typically be done by fording or, say, a bridge. The breath thing is a red herring.
As repeatedly mentioned above: running water is thought to be pure in a variety of religions (probably caused by the fact fast running water, unlike stagnent water, is less likely to hold diseases or bacteria that cause them, they are rushed away, though people did not know this then). Vampires are unholy, unclean things, and thus repelled by the clean and pure, such as running water, or whatever the local Holy symbols were (i.e. crosses). Or silver or salt, both sometimes used to purify food/water (less often with silver, but it does have some germicidal properties). So unholy things are typically repelled by these clean things.
A possible cause of this myth: certain wasting diseases commonly thought in less educated times to be the work of a vampires (i.e. consumption) would not be transmitted as readily across running water, so the "vampire" would seem to have no effect across the river, thus obviously could not have crossed it. Stagent rivers would likely caused or increased likelihood of diseases (cholerea, malaria) which may have been misinterpreted as the work of a vampire. I know this is slightly irrelevant to the question.