B737 Captain reported while enroute to their destination coordinating the need for electrical power before shutting down at the gate; this information was not received by the ramp personnel resulting in an unsafe ramp operation.

Narrative

On this flight we dispatched with an inop APU ZZZ1-ZZZ. We had already had one aircraft swap prior to our arrival and this aircraft was late arriving to the Gate starting us with delays to our evening departure to ZZZ. In considering whether to take the aircraft once Maintenance indicated they would be placing the APU on the MEL I called Dispatch who advised they would reach out to ZZZ ops to make sure they had the equipment and personnel to accept the aircraft in its current status. My primary concern was making sure local ops was aware and prepared to provide ground power to the aircraft with the #1 engine still running on arrival. Dispatch sent an ACARS a short time later indicating the station was aware and able to handle the aircraft. On arrival after exiting the Runway I instructed the FO to call Station Ops to again reiterate and caution that we would be arriving at the Gate without an APU and would be leaving the #1 engine running until electrical power was supplied to the aircraft. Ops acknowledged ramp was aware.The #2 engine was shut down as we pulled into the Gate and once stopped I set the parking brake and awaited ramp personnel to move to connect electric. It became very apparent they had no idea we needed to be hooked up. While trying to coordinate using hand signals someone on the ramp crossed in front of and in close proximity to the intake of the running #1 engine. Our beacon was still flashing. At this point I opened the window to communicate by voice and more hand signals that we needed electric. After a few more minutes electric was finally applied to the aircraft and engine shut down.Because of the confusion I queried station ops on the radio to ask if they had received a message from Company dispatch regarding our APU inop status. Ops acknowledged they had. I inquired as to whether that message had been passed to the ramp personnel that we'd need electric applied to the aircraft before shutting down. Ops indicated the message had been relayed. In closing the loop before leaving the aircraft I spoke with the ramp personnel who indicated that they had no idea; and were surprised to hear; our APU was not in service. The lead saw and recognized the danger of the individual crossing in front of the engine and seemed to attribute the occurrence as being part of the confusion as to what needed to occur when we arrived.In short; despite best efforts prior to departing ZZZ1; communication broke down somewhere which could have resulted in serious injury or worse.