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This article is from the soc.culture.australian FAQ, by Stephen Wales with numerous contributions by others.

12.3.16 Does water go down drains in the opposite way?

If you know anything about fluid dynamics than the fact that the
Rossby number for such a tiny vortex is very large, should give you
the answer. In simple terms: a bath-tub vortex is much too small and
too fast for the Earth's rotation too have any effect on it. Planetary
scale mid-latitude lows and tropical cyclones however do spin in
opposite directions as a result of the rotation of the Earth, but
smaller scale vortices like tornadoes and dust devils do not. [SD]

[MJ] Technically, this is in some sense the case. The Coriolis force
due to the earth's rotation will in the absence of all other effects
cause the whirlpool in a bathtub to rotate in the opposite direction
in different hemispheres. In practice, however, the forces involved
are so small that other effects are more important ie you can make the
whirlpool go in either direction if you give it a nudge. As for using
this effect to determine whether a ship is north or south of the
equator, I don't believe a word of it. The effect increases as you go
away from the equator. It is largest at the poles. It is "zero" on the
equator. Anywhere even remotely near the equator, it is much smaller
than in somewhere like Australia. Combined with the fact that a ship
is likely to get buffeted around by the ocean a little, a much larger
effect, the whole idea is ridiculous.

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