A few things to consider with tarps on a dog kennel run....
If you get snow or rain it will weigh a tarp down a lot more than one thinks. Support with angle to have all that wet slide off is important....very important.
Ventilation, ventilation, ventilation. Yes even though it is outside if you tarp the top and sides it needs to breathe. Don't tarp it so tight that air cannot move.
Consider leaving the south side (or your wind free side) open 100%.
If you get high winds it can grab the tarps and lift the entire kennel. Make sure it is very well anchored. It doesn't take a huge wind to do a lot of damage.
Yes a tightly tarped run can get humid. If you leave a side open or are only putting tarps on top it will not affect humidity.