a "P" trap is very basic. A "P" trap has one main purpose to use the water in the P to make a block to stop the sewer smell or methane gas from entering the room. The second is to catch rings and other small valuables.
Now, why the "P" trap goes dry is another question. Something is not right here. If the water does not stay or evaporates that fast, then the angle of the "P" is so that it drains itself. Or, most likely there is not a "P" even there. The later is probably the cause.
The drain's purpose is to drain out water from the basement during high water tables or heavy rains.
Until that time occurs, I would consider placing a rubber mat over the drain to block the vapors and make sure to remove it when it rains.