Take them off the stem. Not rinsing them is also good advice. If they are quite fresh, they will keep amazingly well in the crisper in the frig.---up to a couple of weeks. Actually, I've picked the last of them just after Thanksgiving and had them for Christmas dinner. They are one tough plant; we are still eating them fresh from the garden, even though nights in Maine are now down in the twenties.

And make sure to remove all outer browning leaves. Do not rinse them under water.

You're asking whether or not you should keep them on the stem? Brussel sprouts will start to become flimsy and flabby if you keep them out. The best solution is to take them off of the stem, and store them in the refrigerator under a damp but not wet cloth. They should keep up to a week.