If you really must have session handling in your API then the client would be responsible to handle the session_id and add it to the URL if required. How exactly to handle this would depend on your technology stack. For example Rails defaults to cookies but (if enabled) would also accept a _session_id parameter as part of the URL.
The relevant information normally stored in the cookie together with the session id then must be handled by the server. In Rails you would have to switch session storage from cookie storage to one of the server options like storing it in the database or memcached with the session_id as key.
But you should really think twice if you want to add this feature to your API. Being stateless will keep your API simpler and easier to maintain. If any possible it is preferable to let the client worry about state and send all necessary information with each request. (like using HTTP basic authentication instead of storing a current user in a session).