Whenever there is a temperature difference on either side of a wall, moisture from the warmer side will tend to collect and condense on the wall. For this reason, the facing on insulation should always face the warmer side. This ensures that moisture condenses on the facing material. If you install the insulation with the facing in the opposite direction, moisture can condense inside the insulation material, causing clumping and contributing to the growth of mold. If the mold doesn't grow inside of the insulation itself, it may grow on the wallboard, causing an unseen respiratory hazard.
The basement is usually the wettest room in the house, and it is often the coldest. These conditions may trump the need for a moisture barrier between the insulation and the floor, especially if you live in a cold, wet climate. Nevertheless, the colder the basement, the higher the gradient between it and the floor above, and the more the likelihood of condensation. Mitigating against excessive condensation on the floor is the fact that warm air rises and flows from the basement to the room above rather than the other way around.
If your basement is cold and wet, it's important to prevent moist air from rising into the insulation, and the best way to do that is to install it with the facing toward the basement. Condensation will probably collect just under the floor, but it may be minimal if the floor above is carpeted. The facing should also be toward the basement if you plan to cover the ceiling and use the basement for a living space. If your basement is unfinished and relatively warm and dry, however, condensation from the floor above may be the most important consideration, so the vapor barrier should face the ceiling.
It may seem reasonable to install insulation with the facing toward the ceiling and then add a second moisture barrier underneath to protect both sides, but this approach is likely to result in damaged insulation. Condensation will still collect in the insulation, and if it can't breathe, it will clump and lose its insulating value. If you install the insulation with the facing toward the ceiling, use straps or wire mesh to hold the insulation in place. Straps are advisable even if the facing is toward the basement, and you have secured the insulation by stapling the facing to the floor joists.