I dont have personal experience with cleaning, however, Basalt is a dense natural stone that can withstand harsh cleaning chemicals. Here are some steps you can take to remove stains from Basalt kitchen:

1. Prepare a cleaning solution of warm water and mild dish soap.
2. Apply the solution to the stained area and let it sit for 15-20 minutes.
3. Scrub the area with a stiff-bristled brush, being careful not to damage the stone.
4. Rinse the area thoroughly with clean water and dry with a soft cloth.
5. For tougher stains, you can use a mixture of baking soda and water to create a paste, apply it to the stain and let it sit for a few minutes before scrubbing and rinsing as above.
6. You can also use a commercial stone cleaner or poultice specifically designed for Basalt stone if the above methods dont work.
7. To prevent future staining, make sure to wipe up spills and splatters as soon as they happen, and regularly seal your Basalt countertop with a stone sealer.