api_v1_application_controller.rb

module API
  module V1
    class ApplicationController < ActionController::Base
      before_filter :authenticate
      TOKEN = ENV['my_api_token'].freeze

      protected

      def authenticate
        authenticate_or_request_with_http_token do |token, _|
          token == TOKEN
        end
      end
    end
  end
end

api_v1_my_controller.rb

module API
  module V1
    class MyController < API::V1::ApplicationController
      respond_to :json
      
      def method
        render json: { status: 200 }
      end
    end
  end
end

call_api.sh

curl -X GET http://localhost:3001/api/my_controller/method -H 'Authorization: Token token="123a"'
curl -X POST http://localhost:3001/api/my_controller/method -H 'Authorization: Token token="123a"' -d { key: 'value' }