What is a good wood to use to build a base for a heavy antique table top?
Wood is wood. So long as it's furniture grade lumber, and adequately designed to support the weight of the tabletop, it doesn't much matter if it's pine or mahogany. What I'd do is to select the wood based on how easy it is to work with and how you want it finished - dark or light; painted or stained or natural; oiled, shellacked or varnished; with lots of obvious wood grain and/or knots, or a tight, even grain, etc. More interesting and aesthetically pleasing wood tends to be more expensive.

Also, you probably want to hire a cabinet maker rather than a carpenter. You don't necessarily need the skills of a furniture maker for this project, but you will need a different set of skills than a typical carpenter has. A good cabinet maker will work with you to match the wood with your design goals.

Finally, maybe take the table as it is to a furniture restorer. It could be the table is substantial enough, it just needs some long-overdue maintenance to tighten the joins and refinish the wood - this will be less expensive than rebuilding it from scratch, and can preserve any value the table has as vintage furniture.