I'm sorry you have to raise your coleus in the basement, but it's best to accept whatever we need to deal with and move on, I guess.
Yes, I like the idea of white boxes - definitely grow lights.
You might want to think about asking somebody who has a sun porch to invite the coleus over for a few weeks, then you can collect them when it's time to put them back in the ground.
I was pleasantly surprised to find, over the course of the last two years, that it IS possible to keep the coleus alive in the house. I have a lot of it - it starts lagging sometimes, but I just persevere with it.
Then come May, out to the table garden it goes - then it starts
growing nicely again.
I just did up two of my pots - one was getting so tall the stalks were sagging - I cut it back and dipped the clippings in hormone, then plugged them back into the soil.
I was sorry to see that tall one start lagging, but it was amazing it did what it did.