IMO, I would move the brew to the stable temperature area ASAP. Depending on where it is in the process will determine any risks of other things happening. IF you just started the batch, then I would get it down there now. Let it ferment, and then rest for a while in the basement. IF there's no way for you to setup a table to bottle in the basement, then simply carefully carry the fermenters back up come bottling day (or a day before). Let them settle before racking into the bottling bucket and proceed as normal. I would then pu the bottles into the basement to carbonate/bottle condition. Unless you are kegging and have the setup in your apartment.
For the summer time, I would just put the fermenters into the basement, then pitch the yeast in. Once done, either bottle down there, or move the brew up to your area as already mentioned.
If you don't want to move the brew up/down 4 flights of stairs, then get/make a fermentation chamber in your place. Get one large enough to hold three (or four) primaries so that you're good to go.
I'm fortunate in that I have a brew-buddy that has a basement that I can place my fermenters in. We brew often enough, and I can even place brews I've made elsewhere (such as a sponsored brew day at the LHBS) to ferment. We try to brew every other weekend, so it's not like I'm leaving something there for a long time. I'll be using the first of my larger SS fermenters there this weekend (25L keg)... It's going to make things even easier for both of us. I've been using 1/6 barrel kegs lately there. Which is still better, IMO, than buckets or carboys. But the new ones are shorter (larger diameter) so they'll be easier to carry, as well as more stable. Almost can't wait to use my pony keg fermenter.