How do I add a new class to an element dynamically?

Short answer no :)

But you could just use the same CSS for the hover like so:

a:hover, .hoverclass {
    background:red;
}

Maybe if you explain why you need the class added, there may be a better solution?

CSS really doesn't have the ability to modify an object in the same manner as JavaScript, so in short - no.

Since everyone has given you jQuery/JS answers to this, I will provide an additional solution. The answer to your question is still no, but using LESS (a CSS Pre-processor) you can do this easily.

.first-class {
  background-color: yellow;
}
.second-class:hover {
  .first-class;
}

Quite simply, any time you hover over .second-class it will give it all the properties of .first-class. Note that it won't add the class permanently, just on hover. You can learn more about LESS here: Getting Started with LESS

Here is a SASS way to do it as well:

.first-class {
  background-color: yellow;
}
.second-class {
  &:hover {
    @extend .first-class;
  }
}