Generally speaking, rain falls when water vapor meets cold air. Typically, the mechanism for humid air meeting cold air is when mountains deflect the humid wind into the cold atmosphere above.
I would guess there are many places in the world which have low rainfall despite having a reasonable level
of humidity. I would also guess these places would be flat areas next to the ocean. The ocean provides the humid air, but the flat terrain means the humid air travels right over without precipitating rain to the land below.
One way of increasing the rainfall in these places would be to build large artificial mountains. But this would be prohibitively expensive. Instead, I propose evaporative cooling.
This would involve spraying a water mist from the top of high towers (perhaps 1km high). The mist would disperse from the towers and eventually evaporate. The evaporation would drop the air temperature and increase the humidity.
The evaporating mist would form a catalyst for a positive feedback loop: mixing the evaporatively cooled air with the humid air (blowing in from the ocean) and causing precipitation, which would then lower the temperature of the area and cause more precipitation.
Apart from acting as an evaporative cooling means, the mist would also help the process by reflecting sunlight and thus lowering the temperature below.
Hopefully the amount of evaporative cooling necessary would be relatively small (maybe a few degrees) to initiate and sustain the process.
Several of the parameters would need to be optimised to make this work. For example, the mist droplets would need to be small enough to be dispersed widely, but big enough so that they don't evaporate too quickly.
Who would fund this? desert comes pretty cheap, so an entrepreneur might be able to create valuable real estate out of nothing, which should cover the intial investment and running costs and leave a tidy profit.