I have 2 bathrooms on the main floor, when I run water in one sink the other bathroom sink starts backing up. What do I need to do to fix this?

Are the sinks back to back? If so, this can be tricky if the right fittings were not used during rough in. You need to get a snake down the drain to clear the blockage. You will want to make sure the snake goes down, watching to make sure it doesn't just go straight over to the other sink. What type of drain pipes are going into the wall? Pvc, or galvenized.

The pipes are galvanized.The bathrooms are side by side.

The pipes are galvanized.The bathrooms are side by side.

This makes me thing that there's a common vent for the two bathrooms.
Snake the lavatory vent. Put out enough cable to reach the base and 18 feet more. Good luck, Tom

Do I run the snake through the drain pipe in the bathroom or do you mean the sewer access in the basement?

Do I run the snake through the drain pipe in the bathroom or do you mean the sewer access in the basement?

Neither one! I mean the lavatory roof vent. One vent should be serving both lavs. Good luck, Tom