The coldest fermentation I've started so far got going at 60 F. It warmed itself up to 66 after a couple of days though. How cold is your basement?
Check out the yeast table (there's one on the main site under "making mead" I think) or go to a yeast manufacturer's website (Lallemand/Lalvin for example) to find temperature ranges for each strain of yeast.
Comparing two very similar cysers I've made with D-47 yeast, one in July and one in December, I can say this so far:
the cooler temperatures made fermentation go slower (but not by much, just a couple of extra days)
the alcohol flavor was less apparent in the cooler fermentation compared to the warmer fermentation (after ~2 weeks).
You can find yeast that will work down into the 40s, and some that don't do well until above 80. Keep in mind that once fermentation starts, it will generate its own heat so you can ferment several degrees above ambient, especially with good thermal shielding (a blanket etc).
Edit: I checked the yeast table available from the main page and the yeasts with the coolest temps listed are the Champagne strains, EC-1118 and Premier Cuvee, which supposedly work down to 45 F. D-47 has a lower limit of 50 listed, for example. Most important for working with low temps is to make sure the yeast are "happy" ("sad" yeast will work ok at high temps but will go to sleep at low temps). Happy means plenty of nutrients, including oxygen to start, right pH, and a proper initial population.
Also, what temperature is your not-basement? Any reason in particular to use the basement? (just curious)