Recently after showering I went down to the basement to find a large amount of water. I've lived in this house for 6 years now and this has never happened before so needless to say I was surprised and there was no obvious drip or source. Today while my roommate was in the shower I stayed in the basement to find the source. It ended up coming from the washing machine drain pipe (there is no washing machine just the open pipe).it started slow then poured continuously. So I found the source and actually plugged it with a t-shirt which stopped the leak completely. I know this isn't a good permanent fix but helps for now. Is there some sort of cap I can get to fix this problem or is there a clog that needs to be fixed before I would cap the pipe? Also some small pieces of toilet paper came out before I stopped the flow.
FYI:
I live in a city, connected to city sewer, not a septic. The shower is on the second floor and the leak in the basement.
Thanks in advance for any help!

You have clogged pipe down the slope from the washer hook up. Have your sewer snaked. Yes, there is a cap you can purchase to cap off the washer stand pipe - but it won't eliminate your exiting problem.

Good advice, that shirt is just a band aide so fashion a temporary cap with a fernco coupler and a plug, you may need that drain sometime in the future. Look for a clean out cap. Once you find it check to see if it is lower or higher than the top of the stand pipe. If it lower you could have standing water behiond the cap. To test I get a 1/8 sheet metal screw and a 1/8 drill bit. Drill a hole in the cap, if water spirts out stick in the screw to plug it. If no water remove the plug and rod the drain line. The clog could be close or as far as the far side of the city street where you connect to the city sewer

You shouldn't have to cap off anything if you snake from a stack clean out or the washer roof vent. Your clog will be found down stream from the washer station. Does every fixture back up out of the washer pipe? Your clog sounds local to me. But to check farther If you can locate the house cleanout, ( it will be located on rhe street side as a rule) open it up. If it's full of water then snake to the street.
Good luck, Tom