I have a house built in 1962 with an unfinished basement with rough ins for toilet and shower. There is a (3?) inch cast iron sewer line and a 1.5 inch galvanized steel line that vents into the sewer line from the top of a sanitary cross. The bottom of the cross goes into the concrete and from there to city sewer. On one of the horizontal connections of the cross goes to a trap and standpipe for my washing machine, the other horizontal connection was plugged. I removed the plug and put in 34 inch 1.5 inch galvanized steel pipe and elbow waste line for a future bathroom sink. After putting in this new pipe, I had a sinking feeling that some of the water from the washing machine would make it to the other side of the sanitary cross and travel up the new pipe, so I tested it. I was right, there actually was quite of bit of water that would come of the new pipe during the spin cycle. So I think I have only two ways of fixing this problem. First is to slow down the washing machine discharge by having it dump into a basin instead. Second is to put a check valve into the new pipe to prevent the washing machine water flowing up the new pipe.
At this point I am not sure what to do. I know there are similar houses in my neighborhood with basement bathrooms but I don't know how they addressed this problem. Any insight would help.

Subject line should have been: Washing machine discharge flows across sanitary cross into new pipe.

Usually, 2" drain will absorb w/m discharge w/o backing up. If there is obstruction inside 2" pipe than you will encounter water back ups during w/m discharge. Also, if your drain is 1 1/2" than it is undersized and drain overflows are common.

If your cross San T is 2"x 2" x 1 1/2" x 1 1/2" than you shouldn't have overflow problems.

Yes, good option is to drain w/m into service sink. I do not like the check-valve option.