Hi there!
We just moved into our home about 6 months ago. It was built in the 70's, and is really in great condition. When they did the inspection, they did notice a little standing water under the laundry room. We (and the inspector) dismissed it as a problem with the previous owners old washing machine.
Starting about 2 months ago, we started having a VERY strong sewer smell coming from the laundry room. We run the washer and dryer every day, so the smell is consistent.

We tried taping off where the drain pipe goes into the stand pipe- It has not helped.

Any advice you have would be wonderful!

Do you still see water on the floor like during the inspection? Are there any floor drains in this room or other fixtures with drains? My first guess is that you have a untrapped drain somewhere but it is too early to make such a guess. Do you know where your vent stack is located? Come back with your answers.