Why did Joy give Sadness the core memories?

Early on in the movie the other emotions ostracized sadness because she only seems to get in the way and slow Riley down. They don't understand why someone would need sadness in their life.

Later on in the movie, while Joy and Sadness are working their way through Riley's mind, trying to get back to the control center, Joy is witness to Sadness's importance. She sees how Sadness can actually make someone feel better by addressing what makes them sad.

Once they return to the control center Joy has matured and sees that the best way to help Riley contend with the move and adjust to the new life isn't to cover if up in Joy, but to let her feel the Sadness so she can begin the healing process and grow up. This is why Joy allows Sadness to interact with the core memories.

Growing up and dealing with complicated, adult emotions, is the major theme of the movie.

I agree with onewho: Joy realised that Sadness is a vital part of emotional balance of any grown up.

In addition: While we want children to live an all-happy life, we require adults to be able to handle (instead of ignore) whatever sadness they have to bear.

But probably the most important part is not the addition of the all-sad memories, but of those mixed-color ones.

Children tend to have simple emotions. Joy, sadness, one at a time. As we grow up, we have way more complex emotions. Think of parents whose child moves into it´s first own appartment: They are proud, happy, but also sad at the same time.

These mixture of emotions - their cooperations and simultanious presence - is symbolized by the bigger control panel, where 5 people can stand at once.