The weakness that prevents vampires crossing running water is derived from Bram Stoker's Dracula, but found only in D&D 3.0 and 3.5.
Advice from AD&D suggests that vampires should charm people and use them to circumvent weaknesses.
What is the history of the D&D Vampire's "running water" weakness?
The D&D vampire seems to be based on Bram Stoker's Dracula, in which it was noted that "The Count, even if he takes the form of a bat, cannot cross the running water of his own volition" (Chapter 25, Harker's Journal, 15 October). Van Helsing also explained, "It is said he can only pass running water at the slack or the flood of the tide" (Chapter 18, Mina Harker's Journal, 30 September).
The earliest editions of D&D included vampires, but did not mention this limitation. For example, the OD&D Monsters & Treasure volume (1974, p.10) noted only these weaknesses:
Vampires cannot abide the smell of garlic, the face of a mirror, or the sight of cross.
They will fall back from these if strongly presented. They must always return to
a coffin whose bottom is covered with soil from their native land during the daylight hours.
(A partial quotation can be found here.) These are drawn from Stoker's book, but no mention of running water.
As far as I can tell, the earliest mention of it was in the 1st edition AD&D Monster Manual (p.99; also quoted here), first published in 1977, which included the following:
A vampire can be slain by the following methods: exposure to direct sunlight kills the creature in 1 turn, and it becomes powerless immediately. A vampire immersed in running water for 3 melee rounds is killed. If a wooden stake is driven through a vampire’s heart it is killed, but only for so long as the stake remains; to finish this task the vampire’s head must also be cut off and its mouth filled with holy wafers. (emphasis added)
The Basic D&D rules, also published in 1977, contain a similar description:
Vampires can only be killed by being impaled through the heart with a wooden stick, exposed to direct sunlight or immersed in running water. (emphasis added)
The preface to that book says it was based on the original rules as well as intervening supplements, but I can't find any that references running water. So it seems that the rule was added in Basic and Advanced D&D in 1977.
The AD&D Ravenloft setting naturally required far more detailed vampire lore. This was provided in a book called Van Richten's Guide to Vampires (1991; reproduced here), which seemed to provide an explanation as to why the D&D vampire did not match Stoker's (pp.37-38):
Many legends tell of vampires being kept at bay by running water, and conclude that running water somehow has some warding power over vampires as does garlic and mirrors. As far as I can tell, these tales are probably true with regard to the base events, but totally wrong in their conclusions. It is true that all but the most powerful vampires - generally speaking, Eminents and Patriarchs are quickly destroyed if they are immersed in running water. Vampires are, of course, aware of this vulnerability, and hence will avoid running water if there is a chance they can be immersed in it. This means that vampires will be particularly wary of bridges, stepping stones, ferries, and other means of crossing running water. (After all, bridges can collapse, ferries can sink, etc.) If the benefit is great enough, vampires will risk such means of crossing running water but will always do whatever it takes to minimize the risk. If circumstances allow, however, the fiends will shapechange to bat form and fly across a river. Thus it can be seen that an aversion to crossing running water is not a strict prohibition, but merely a rational choice.
There is one exception: a vampire in gaseous form is strictly prohibited from crossing a body of running water that is more than three feet wide.
Ravenloft also modifies vampires so that they are not killed by immersion in running water, but simply involuntarily transformed into gaseous form and forced to immediately return to their sanctuary (p.45).
It wasn't until D&D 3.0's Monster Manual, published in 2000, that the rule about death by immersion was supplemented by Stoker's inability to cross running water. The rules for creating a vampire stated:
Repelling a vampire: … Vampires are also unable to cross running water, although they can be carried over it while resting in their coffins or aboard a ship. (emphasis added)
Slaying a vampire: … Similarly, immersing a vampire in running water robs it of one-third of its hit points each round until it is destroyed at the end of the third round [of immersion].
This wording was retained by the 3.5 Monster Manual (2003, p.253), with the words "of immersion" added as indicated above.
Interestingly, in 4th Edition, this rule was rolled back, along with vampire weaknesses. The Monster Manual (2008, p.258) provided:
A character knows the following information with a successful Religion check.
DC 15: Contrary to popular folklore, vampires are not hampered by running water or repelled by garlic, and they don’t need invitations to enter homes. (emphasis added)
(It is interesting that the rule was written in this way. It seems like a kind of "easter egg" for DMs whose players are used to the 3.5 rules, and may mistakenly believe the "popular folklore" about running water. Well played, WoTC.)
How should a DM run a Vampire who can't cross running water?
It is interesting that the 3.0 and 3.5 books, which are the only editions that contain the restriction on vampires crossing running water, don't provide any clues as to how this would be countered by a vampire.
However, AD&D provides some clues. For instance, the AD&D 2nd Edition Monstrous Manual (1993, p.355) makes this comment:
In most cases, the vampire will seek to overcome these hazards with the aid of its minions, For example, a charmed person might be called upon to attack someone who is holding the vampire at bay with a holy symbol.
And later (p.356):
In addition to its aversion to items like garlic and holy symbols, the vampire acts under many other limitations. One of the most powerful of these is the creature's inability to enter a home without being first invited to do so by a resident of the dwelling. … A common manner for obtaining permission to enter a home is the use of the vampire's gaze to charm a servant or other inhabitant.
This suggests that the strategy adopted by your DM is consistent with the concept of the AD&D 2e vampire.