When a Horsea with the ability Swift Swim evolves, Seadra will have Poison Point.
So, yes, the ability does change after evolution.
The only way a Pokémon can change its Ability is upon evolution. The "slot" storage system means that if an unevolved Pokémon has the "first" Ability for its species, it will have the "first" Ability of its evolved species upon reaching this stage, even if this Ability differs in the evolved species. In other words, a Poochyena with Run Away would evolve into a Mightyena with Intimidate, while a Poochyena with Quick Feet would evolve into a Mightyena with Quick Feet.
Ability Mechanics, Bulbapedia
And Horsea's first ability is Swift Swim. Therefore when it evolves it will have the first ability of Seadra, which is Poison Point.