The Stasis Chamber riddle is solver near the end of the game, once you reach the location called Broken Fathom. It happens during the main quest The Right to Exist. Once you solve the riddle, you will be able to enter the Stasis Chamber and examine it. You will receive EXP for that. After the conversation with Ghostly Woman you will be also able to receive free healing.
The above picture will help you in solving the riddle. The path to the central Stasis Chamber can be unlocked by interacting with the nearby control panels (points 1-5 on the map). Below is a list of steps you must complete:
1) Interact with panel 1 and activate it.
2) Walk to panel 2 and establish connection with panel 3.
3) Walk to panel 3 and select the first option from the list - pressing the pulsing picture with a human drawing. All panels should become highlighted.
4) Leave panel 3 and go back to panel 2.
5) Interact with panel 2 and select the second option - pressing the button with human drawing. A clone of the main character will appear.
6) Speak to the clone, use all dialogue options and ask him for favor. The clone will walk to panel 3 he will use it.
7) Clone's action will lead to the appearance of a passage to the Stasis Chamber.
Once you complete all the actions, you can reach the portal near the panel 5 and use it to leave the location.