How does convection affect ocean currents?

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In fact very little related to mantel convection, but quite a bit related to atmospheric convection.

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If by convection you mean the currents within the Earth's "mantle" the answer is : not at all. However, atmospheric convection cells are a major driver of ocean currents. These big atmospheric convection cells create surface winds which in turn set up ocean current movements. See pic.

https://en.wikipedia.org/wiki/Hadley_cell image source here

If you intend whatever convection may be generated by thermal expansion within the ocean mass, this effect is to be seen in combination with a stronger dynamism behind the ocean currents. The one generated by the planet's rotation about its axis.

This force, known as "Coriolis" after Gaspard-Gustave Coriolis (1792-1843) the French Geophysicist who first pointed out its effect, imparts a superficial rotational movement to all fluid masses on a spinning body.
Check the attached map from Wikipedia. The main currents, in red, all move clockwise in the Northern hemisphere. They move anti-clockwise everywhere in the Southern hemisphere.

enter image source here

On the other hand, the blue currents run in the opposite directions. These however move at depth, close to the ocean floor, and are due to the cold water restoring the displaced volumes of the warm water masses.

What in fact happens is that equatorial temperatures expand the ocean water forcing it to move beyond its constrains. At this point, the Coriolis force takes over and gives the current its rotational direction.

Have you heard the weird story of the whirl of an emptying bathtub turning one way or another in North and South America?
Don't you believe it. It is not true. Coriolis cannot have a measurable effect at that scale.

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