What you have doesn't sound like any kind of catch basin I've ever seen, nor is it an ejector pump or sump pump or, obviously -- due to the lack of a pump -- any kind of pump. If it is essentially dry in the "basin" between uses, I would guess that it's some type of laundry-tub for the water to go into until it can drain out of the house. (By the way, another clue to the "pump" issue is the lack of a pipe going UP out of the basin into piping that would be located at least somewhat above it. A catch basin's purpose is to literally "catch" soap and fats from pipes feeding into it from sinks all over the house, your bathtub and washing machine and washtubs. Often, drain lines come into it from pipes that connect it to your downspouts. The soaps and fats float in a constant pool of water in the catch basin and generally form a thickened ball or layer on top that should be removed every year or so. When the catch basin gets too nasty, there are companies that come out with tanks and suck out the contents. The better ones also scrub down the walls and add some lime as a deodorizer. Also, since many catch basins are brick, at this time it is good to have someone (hopefully the same service) check to see if any tuckpointing or brick replacement needs to be done. Oh.... The water exits the catch basin -- via gravity when the water reaches a certain level -- thru a pipe near the bottom of the basin. A catch basin usually also has a heavy sewer-cover-type lid. It's easy to check which sinks, etc., feed into it by simply having one person watch the pipe that comes into the basin and another person go and pour/draining several gallons of water down each sink or other draining fixture and see which ones result in water entering the basin. The toilet should definitely NOT drain thru the catch basin. If it does, contact a plumber right away as this is a highly unsanitary situation.
Most people have their laundry tubs drain out thru the same plumbing lines that their sinks and bathtub go into. I'd guess that the main reason for your odor is that you have an open permanently damp line going to the outside. If a trap can be added to it, that could just about end the odor issue. Water would stay in the trap and prevent the pipe from venting back into your basement. If you have any floor drains, you can redirect the water from your washer out thru a floor drain, I don't know how effective that would be long term. Or, as I mentioned at first if you have any wastewater pipes that go thru your basement that would be the ideal solution, drain straight into sewer system and call it a day. (The wastewater line would have to be at least a little lower than the water outlets from your sink and washer.)
The cheapest, easiest solution (if I'm right about what you're looking at there) is to add a normal washtub with legs. Have a regular trap coming off the drain of the washtub and run PVC piping from there and out of the house (make sure you have a slight downhill slope on the pipe). I'd guess you could run it out straight thru where the pipe runs now. The outside end has to either come out into the air (put a screen over the end so you don't have visitors using it to come into your house); OR, dig a dry well outside the house somewhere (basically a pit full of gravel a couple of feet deep or so and under 6 inches to a foot of soil. Make sure to leave a note to yourself, wherever you keep your other maintenance files, about where it is, or you'll never find it again; Or you could run your pipe out into your draintile or bubble pot.
I know that this is probably more information than you wanted, but I don't have your house to look at so I wanted to cover the bases. Oh, lastly, anybody that does have an actual catch basin, using those packets once a month that they sell for septic systems sets up a good bacterial colony and helps eat up some of the crud and then, when it gets a bit stinky, put lime in your basin in the correct amount for your size/style of catch basin (ask either a plumber or somebody who actually knows something about these systems). Good luck !