They can cross over (by flight, bridge, or being carried) running water, but not pass THROUGH the water (swimming, etc) lest they risk permanent destruction, as their mist form escape doesn't function when they are slain by running water.
Classic Horrors Revisited very briefly mentions them being thwarted by running water, but doesn't specify they can't cross it, so I'm assuming that's a reference to the damage they take if immersed. It also very specifically says that many vampires don't adhere to the generic vampire rules, as I point out below, and relying on a single "traditional" weakness could be an adventurer's last mistake.
At the table
While your question asks for RAW, with all our talk of vampires (now spanning three questions), I feel I should mention that vampires are one of the most, if not the most, house-ruled monsters there are. Adding and subtracting classic weaknesses, separating them into clans (along lines such as games like TES4:Oblivion or Vampire: the Masquerade), or simply giving them class levels you might not expect (It's a vampire... cleric of a neutral deity? Channeling negative, oh dear...); all of these and more are ways vampires quickly find themselves telling RAW to bite them (pardon the terrible pun). If your vampire friend finds himself stymied by running water, great; beware your DM didn't remove a different classic weakness to add that one. If the vampire is in full-plate, don't assume that means he can't cast fireball; the Tremere clan's blood magic cares not for your arcane-spell-failure rules.