Persistent sewage smell coming from basement drain
You are correct that floor drains do need to be monitored and occasionally the trap refilled with water.

So the first thing to check is that your floor drain actually has a trap. The way to do that is to slowly pour water in to the drain. You should notice the water level rise and stay there. If the water disappears quickly then it's likely you don't have a trap.

Provided that the drain does have a trap, the next thing to figure out is why the trap is emptying so quickly. A common cause is something wrong with your vent stack. Problems can range from a blockage, to an improperly vented drain elsewhere, to no venting at all.

A good way to test your vent stack is to flush each toilet one at a time. Enlist a helper and keep an eye on the water in the trap. If one toilet causes the water in the trap to move significantly or drain completely, you have found your problem fixture or vent stack. Fixing this could be as simple as running a garden hose up to your roof and down that vent stack to clear the blockage.

If climbing on the roof is not for you, then a standard house-call charge from a plumber will likely fix it, and that's not usually too expensive.

But if a fixture is incorrectly plumbed or inadequately vented, that can get pricey.