The answer is "Yes and no!"
There IS an influence, called the Coriolis Effect, which affects LARGE systems, such as weather and sea-currents moving over the earth's surface. However, it has no real influence on something as relatively tiny as a basinful/bathful of water. In fact, the eddies and currents created by filling the bath are vastly more powerful and would take days to settle down to complete stillness. Consequently, the water will generally flow OUT in the same direction as that in which it flowed IN. If your hot tap is on the right and your bath has mostly hot water in it, inflowing water will largely circle clockwise and vice versa if the tap-positions are reversed. Consequently, that will almost certainly be the direction in which the water will flow OUT, too. So, there - re small containers - is the "No" answer
Engineers at the Massachussetts Institute of Technology - clearly in the northern hemisphere - built a large, wide tank in the 1950s with a small plug-hole at its centre. They filled it and left it to settle for several days, then opened the valve. In each of the many repetitions of the experiment, the water DID start to swirl in an anticlockwise direction. When they deliberately set up a clockwise swirl, it invariably died down and returned to anticlockwise. The explanation, apparently is that - if the water has settled completely - it is rotating, along with everything else on earth. The fluid particles nearest to the equator are obviously moving more quickly than those furthest from it and, when they are released through the plug-hole, they develop their characteristic direction-finding properties.