Ants only appear when they have a food source, such as aphids. Get rid of the food source and you get rid of the ants. Yes safer's soap will kill the aphids. Most soaps also kill most ants right on contact.
I suggest adding a strong smelling essential oil to the soap spray. Ants navigate by putting down chemical trails that they follow. Throw off their scent trails and they are totally lost and disoriented.
You can also use ant bait traps to get the ones in the perimeters. Find out first what kind of ants you have. Roughly 2 types of traps, one for ants that go after sugars, and one for ants that go after oils.
Hope this helps.
Oscar, I have an organic clove and peppermint oil based insecticide that I like, but I'm afraid of burning tender seedlings. Any idea how much I could safely add to my safer soap spray?
Thanks for the link, Ethan. I agree with your Borg comparison. They are relentless, but I refuse to accept that resistance is futile!