W few years ago we decided to clean out the rocks under a huge basswood tree, and ever since then it was sending up many suckers from the base of the tree. How do I prevent this from happening?
You answered your question yourself. You cleared out the base of the tree. The tree has the evolutionary instinct to reproduce. Therefore, it is putting out suckers. Without covering up the base in some manner, you will never be able to stop the suckers.
You can buy and spray some herbicides/weed killers from your local gardening store. You will not always have success and you may end up harming your tree for very little in return. You should research and make sure whatever you spray is not going to harm your specific tree. Following this, you will still need to restrict future growth somehow, anyway, so I'd forego this and just weed by hand.
You can put down barrier cloths, which are no different from the idea with the rocks. So, you can choose to do something permanent, or, if you like doing things by hand, you could just go down once every so often and give them a good weeding.
Lastly, some trees take very unkindly to pruning and will put out shoots with a vengeance, so find out the right season where growth is about to slow down, like Fall, to apply growth restrictions such as plastic tarps/cloths, rocks, etc.