The sunlight radiation is very important for the climate, so I understand the answer that is saying "no change, because the sun goes through". However, I don't believe this is quite right.
Your alien will need energy. No way out of this. They have computer/fridges/whatever, so they need to power it. If they don't get it from the sun, they need to get it from the planet. Let's consider a few options:
Ok, you said they won't. But the easiest way for them would be to take some solar energy. Even if they plant trees to feed themselves, trees are actually taking sun energy. And they are shading the earth below. If the sun goes 100% through their city, it means they are 100% transparent (the city and themselves). Highly unlikely.
If you can see them or their cities, they are blocking some solar radiation.
If they have vegetables, trees or solar panels, they are already using solar energy.
Ok, let's assume they didn't touch the sunlight. Everything is transparent in their cities. The sun goes 100% to the ground.
So the first and most obvious energy source would be the wind. However, slowing the wind is not without effect. (Yes, if you take energy from it, you are slowing it). See for instance this article.
Actually, even if they don't take energy from it, it is likely that their cities in the sky are going to affect the wind.
Basically you can count on poorer energy redistribution. Warm places are going to get warmer. Cold places colder.
They could condensate the water up there, directly from the cloud, and use it as it used in dams. So let it drop through their pillars and put a turbine at the bottom of it.
The effect would be to reduce the amount of rain on much of the planet. All the water would go down through pillars, no through rain. Everywhere would turn into a desert. Only exception would be the deserts, as you said the pillars are in there. So the area around pillar basis would get too much water, the rest would be dry.
Also consider the effet of releasing a lot of non-salty water in oceans. Likely to change ocean currents, like stopping the gulf stream.
For instance stealing petrol or uranium from us. Will likely result in pollution and wars. Catastrophic for the climate. Just as we know it these days, just worse.