Did you find out where the water was getting in on the front wall that you replaced? You did not mention the roof, and if the problem was up above, your new wall will get moisture in it. You should not have to use treated lumber except for the bottom plate, and if water gets inside the wall, even if the lumber is treated and won't rot, it will cause other problems. So you have to fix the source.

If water is coming in around one of the old windows on the other wall, you should first dismount the window and check the seal around it. This will require removing siding around it on the outside of the wall. If it was not installed or caulked properly, you may be able to remedy that. Unless that wall is rotted too.

As for the condensation, you may be right and some ventilation is needed. A small vent may do the trick, with an angled metal louver inside and out so rain doesn't get in.

Condensation on a window is caused by warmer air in a space contacting a colder structure, wall, door, window, mirror etc. No reason whatsoever to ventilate, might as well leave the window open a crack to lower the breezeway temperature. Yes the leaking room helped with the condensation as the internal and external differential temperature were less than now. It is warmer in the breezeway now, for certain.

That is why newer windows are of higher R values helps keep condensation down,glass will still fog up if the difference between the interior temp of the room, car, etc with higher relative humidity is great enough compared to the interior temp of the glass. Thus turning the water as a gas into it's liquid form.

You may experience moisture on the window of the north wall fogging up more than the south window as that north exposure will be colder than the south. That is why in the woods we can tell north from south as the north side of a tree is darker than the south side of the tree. It simply sees less sunlight.

First step is to install the windows properly and seal that leak. Remove the window and flash it properly. Spend a few dollars and replace the single pane storm with a double pane insulated window. Seal it using a low expansion foam sealant, there are foam sealants out there that cause more problems and can actually cause the window to leak. Some sealants contract over time creating gaps and some expand so greatly they will actually push the frame out of shape. Use a sealant specific to windows and doors. Replacing with an insulated window WILL stop the condensation for an unheated room, it will completely STOP the problem unless you use this breezeway to boil water in it.

Where exactly is the water seen leaking in ? Above the window, below the window, beside it? Just because the leak shows at the window doesn't mean that's where the leak is. Water can migrate in the strangest of ways. I have found leaks many feet away from where it first shows itself. Once had a roof leak show on a first floor wall but the walls and ceiling on the floor directly below it was dry as a bone.

Find the source of the leak first. I want you to replace the window with an insulated window regardless if the leak is from the window. However if replacing the storm window is not in the budget, which is certainly understandable, not all is lost. You can run a dehumidifier as a temporary fix until you replace the storm windows. Humidity should be between 50% and 60% in a space. Heck you might be in south FLA. Where humidity is an issue year long, usually adding humidity in the winter around here. Dehumidifier will stop the problem as well but it should be temporary. The money you will spend on your power bill will eventually be costing you more than the cost of new windows. With good windows you should be able to heat that room and not have moisture on the windows. Remember when I was a kid the windows in the kitchen were always fogged up, but not in the rest of the house. Which begs the question why is that room so damp? Is it on a slab, is it built directly on grade without a moisture barrier, is there a vent from the home that terminates in the ceiling from a bathroom or kitchen that originally vented to the exterior before the breezeway was built. I have seen it many times before.

A friend of a friend built a new home this past summer and complained all the kitchen windows were fogging up now that winter is here. He cheaped out on windows, they were a steal he said, and his wife had the slow cooker going on the countertop close to the windows, and drinks tea continuously loading that kitchen with moisture. Moved the slow cooker, turned on the vent fan above the stove when cooking and problem solved, they still have windows that are going to cost him valuable hard earned coin over the years with heat loss. Pay me now or pay me latter as the saying goes.

Hope this helps.

While it's still on my mind, Toxcrusadr suggests removing siding on the exterior wall to replace a window, and or check for leaks, for the life of me I can't imagine why. That is assuming there is siding to remove.

Toxcrusadr, I may be missing something but why would we do that? Never had to before, only have if I wanted to replace flashing or add flashing if none exists and couldn't tuck it below the siding. Sure would be tough if there was brick vanier (sic), spellcheck just died on Windows8RT again, or wood clapboard.

I have a leak test procedure using water when the source isn't obvious, not the best method in the winter up here or unless the it has rained and everything needs to be bone dry.