Inspect the side accordion panels on the window unit. These panels are supposed to extend from the unit to the sides of the windows. If there is a gap, rain may leak in. Loosen the screws around the panels and stretch them out wider to tighten any gaps. Tighten the screws again.
Check for holes in the accordion panels and in the seal around the window frame itself. If you see holes, seal them with caulk.
Ensure the window unit is level. A tilt in the unit may make moisture run inside the window. If the unit is tilted, loosen the screws around the accordion panels, level the unit and tighten the screws again.
Install foam weather stripping around the unit and the window to keep moisture out.
Erect a canopy over the unit to prevent rain from reaching the unit or the window. Awnings and independent tent structures are options for this type of task.