Since you won't play PE I will. :laughing:
You did not mention the span of the joists. A simple solution
could just be adding a beam under the joists with the heavy load and putting a post(s) under said beam. Course the unsimple part is what the post(s) is resting on handle the weight and what size post(s) do you need? And should the post be metal and not wood. Which is cheaper? Will the post(s) be in the way in the basement?
A coworker years back was having a foundation and floor problem in a new house. He paid a couple hundred dollars max for a PE to look at the problem and provide a design for a fix. The fix was a 4x4 in the crawl space under the floor joists with a proper footer. In your case I would be concerned about a 4x4 working due to the height of the basement and/or you will need more than one. A metal post I would bet could handle the load cheaper than a few wood posts.
A load baring beam and post(s) could dramatically change the load capacity of the floor. But will it be enough? And which would be cheaper, beam/posts or beefing up the joists? Mr. PE would know and save time and effort. This really depends on the length of joist span, the joist size and where the load will be placed on the joists.
I bought the NDS, National Design Specification, years ago when designing our house. I AM DANGEROUS.



But since I KNOW I am dangerous I used a PE to build our house.
Later,
Dan