If the washer does not agitate at all, does not spin, but will still drain, this is a symptom of a broken motor coupling.
The following link explains how to replace one:
This is a very simple repair
that doesn't cost much. The part number is included in the link I provided. Appliancepartspros.com carries a replacement motor coupling for about $11.
NOTE: A broken motor coupling is usually accompanied by a loud rattling noise. You can hear the motor engaging and running, but there is no tub action.
If you need an illustration, you can refer to the website I mentioned and locate item 13 under the Brake/Clutch/Gearcase/Motor/Pump section.
If you have any questions about this repair, or if you need additional assistance, please let me know. I hope you find this information helpful.