Spells Don't Cross Planes, Unless Specified
So, banishment cast from the material plane on a character on the ethereal plane would fail.
A traveler on the Ethereal Plane is invisible and utterly silent to someone on the overlapped plane, and solid objects on the overlapped plane don't hamper the movement of a creature in the Border Ethereal. The exceptions are certain magical effects (including anything made of magical force) and living beings. (DMG p. 48)
Spells do Exactly what they Say
If the creature casting banishment is on the Border Ethereal also, then banishment will do what it says:
You attempt to send one creature that you can see within range to another plane of existence. The target must succeed on a Charisma saving throw or be banished.
If the target is native to a different plane of existence than the one you're on, the target is banished with a faint popping noise, returning to its home plane. If the spell ends before 1 minute has passed, the target reappears in the space it left or in the nearest unoccupied space if that space is occupied. Otherwise, the target doesn't return.
Which because it was cast last, would surpress the effects of the etherealness. If the banishment ends earlier than a minute and the etherealness duration hasn't run out, then the character would return the Border Ethereal plane until the etherealness ends. If the full minute of banishment happens, the character remains banished, as the spell says.
Just Like Summoning
Note, this is no different than if you banished a summoned creature. Take a look at the wording of summon greater demon or conjure woodland beings.
The demon appears in an unoccupied space you can see within range, and the demon disappears when it drops to 0 hit points or when the spell ends.
A summoned creature disappears when it drops to 0 hit points or when the spell ends.
Neither mention being banished, and the only way it mentions is dropping to zero or the spell ending; but clearly banishment or wish or planeshift would work to also make them disappear.
The Difference in Language
The difference in language is that etheralness contains the line:
You remain in the Border Ethereal
This is because you are the one going, and you might want to "pop in and out" like a more controlled blink spell; this is saying that isn't what this spell does.