With nobody there, who is going to stop them from climbing the fence. Also, how are you going to fence the middle of the Rio Grande?
People keep asking this question as though it's one of those eternal universal mysteries that can't be solved.
The Rio Grande can be fenced on the US side, as long as you provide controlled access points for those US citizens and agencies that require ready access to the water.
Put more border guard personnel and stronger security measures at those access points, than along the rest of the fence. This just isn't an insoluble problem.