I have water coming up from a drain in my basement when I use the washing Machine. My house was built in 1919, so it could be a number of things. Is this terribly expense to fix. Should I be trying something on my own before calling a plumber? Any advice or ideas are welcome. Thank you

This will require snaking the drain. Does your floor drain(drian that water is backing up into) have a side cleanout? Or does your washer drain have a cleanout? You can snake the drain from either of these places. Make sure to run enough cable to reach the nearest bathroom, plus ten feet(just to make sure you run enough cable.). This is a common occurrence. If you don't feel comfy snaking the drain(can be a VERY messy job) I would suggest calling in a plumber, they will have you fixed up in short order. May cost a bit, but it will be done right, and if they are good plumbers, they will clean up after themselves(unless you want to save some money and tell them that you will clean up afterwards). Please let me know how things turn out, or if you would like more advice on doing this yourself.

After you're through snaking the drain line let's prevent the floor drains from backing up in the future. There's a floor drain check valve called Flood Guard,(see image) check it out at: Flood guards for floor drains - check valve to prevent flooding from FAMOUS PLUMBING SUPPLY
Good luck, Tom