Easiest fix is to change the server side to not authenticate the CSRF token. Here's an example of using a different controller for your API.
class Api::BaseController < ApplicationController
skip_before_filter :verify_authenticity_token
end
In general, your API is either going to require authentication for API calls (in which case you should have your own authentication, or OAuth, or any number of authentication mechanisms) or isn't (in which case it's a publicly accessible API and CSRF doesn't matter). There a few other threads here and here that discuss it.
From another answer on SO (go upvote it!):
CSRF attacks rely on cookies being implicitly sent with all requests to a particular domain. If your API endpoints do not allow cookie-based authentication, you should be good.