Well the easy answers are this:
Light exposure - One assumes that since it's in the basement that won't be a problem.
Temperature - Not only ambient temperature, but whether or not the fermentation vessel will be on the concrete, which will leech heat.
Type of beer you brew. Lagers and saisons aren't going to get made in the basement closet. :-)
Out of sight, out of mind. You will need to aware that you have a beer down there fermenting. I know it sounds like a dumb problem, but everybody has busy lives and if you can't see it, you might forget it. Just something to be aware of.
You will need to get a temperature profile over the year, finding out how low the temperature goes and conversely how high it goes in the summer. You can than compare that to the temperature profiles of popular ale yeasts.
For my part, the basement is a good environment to ferment, store, and age beer. Basements are generally dark, cool environments, making them perfect for this purpose.
Congrats on the new house and good luck!