Short answer: You can have cliffs like this, but they're not going to be as high as the Andes.
Long Answer: You have several options for creating cliffs like this, but none of them are going to be as high as the Andes.
Let's talk about mountain formation
Option one: Fault Block Mountains
Maybe you're thinking of something like this:
...but with mountains. Welcome to the Telascica Cliffs in Kornati National Park, Croatia. Let's zoom out and take a broader look:
In the pics on that page, you can see how it's just the western face of the cliffs that rise dramatically out of the water. The eastern side of the island gently slopes down. That's because the cliffs are a tilted block on a fault line.
You can scale up tilted blocks to the size of mountains, and what you get is the Big Sur:
Another variation of fault block mountains is are horst and graben cliffs. The scaled-up version of these is a rift valley with a narrow sea between them (as kingledion points out, if you have enough time for an ocean to form, beaches will have formed).
In both cases, these cliffs are usually about a mile high before erosion.
Option two: extreme erosion
In this case, you're looking at something like the Na Pali coast.
Whether or not you get this is dependent on what type of rock you have and how it erodes. It's also going to be about a mile high.
Another way to do extreme erosion is fjords.
Now let's talk about the other half of this: beach erosion
The easiest way for you to not have beaches is to have something that gets rid of beaches. These processes exist. To quote this lecture: http://www.iupui.edu/~g115/mod13/lecture05.html
Eroded coastlines are defined by wave energy erosion exceeding sediment deposition.
In other words, you want your waves to carry sand away faster than it can deposit. This can be done with relatively swift offshore and nearshore currents and riptides.
It helps if your mountains are made of relatively hard rock, like gneiss, that erodes slowly compared to how quickly the ocean washes sand away. The west coast of the US also has submarine canyons that give the sand a place to go as the current carries it away from the cliffs.
Last but not least...
You might be disappointed about your mountains realistically being a mile high. Keep in mind the Velebit in Croatia are about a mile high.
Also, this is high enough to cause a pressure differential in winter that results in strong winds. Your ocean immediately adjacent to your mountains might not be navigable for part of the year.