Sewer lines need to be vented in order not to create a pressure differential that causes problems. There are a couple different approaches, but that shit isn't one of them; it kind of looks like something a plumber just threw in there to solve their immediate problem and then scooted away from free of consequence.
I wouldn't recommend plugging it, as you might be taking away a needed vent from the whole system... the two options I could see would be:
Hire a plumber to fix the bullshit in more proper fashion
Replace that rubber hose at the end with a longer hose, and put the open end at the bottom of a bucket that you keep filled with water (either in the sink or on the ground). You're effectively creating a weird custom type of P-trap. It's ghetto but it'll solve the smell problem while still allowing pressure to equalize, I think. You might want to ask your plumber if that would be a sensible solution just to double-check.
I’m open to the ghetto solution for the time being while I cap it. I replied to another comment with a pic of the main vent stack so it looks like I’ll be able to cap it. Just have to figure out the best way to do that.
They make what is called a studor vent for this situation. Not sure if there are code restrictions for different areas. If allowed you can put them under sinks etc to get better venting of sewer lines.