Yes, and before framing rather than after. Retrofitting a system into a finished basement means opening drywall to route pipe, which turns a straightforward install into a demolition and repair job.
More detail
Sequencing is the entire point here. In an unfinished basement, a mitigation system is a clean install: core the slab at the chosen suction point, run pipe up an unobstructed wall or through a mechanical chase, mount the fan outside the living envelope, and seal the penetrations. In a finished basement, that same pipe run has to pass through framed walls, insulation, and drywall you just paid for, and the suction point has to be cut through new flooring. There is a second reason to test first. Finishing a basement changes how the space performs. You are adding conditioned, occupied square footage on the lowest level, which is where radon concentrates, and you are typically tightening the envelope and changing air movement in the process. A basement that tested acceptably as unfinished storage is being converted into the room where people spend hours a day. The practical sequence: test before design. If the result is above 4.0 pCi/L, install the system before framing so the pipe route is chosen while walls are open and the chase can be planned into the layout rather than boxed in awkwardly afterward. If the result is between 2.0 and 4.0, it is worth stubbing in a rough-in pipe during construction even if you do not install a fan, because adding a fan to an existing pipe later is inexpensive and cutting in a pipe later is not. The same logic applies to basement waterproofing work, which also opens the slab perimeter. Coordinating the two is cheaper than sequencing them apart. Basement finishing is common across Greater Columbus and Central Ohio and so is the sequencing mistake. Permit requirements for the electrical portion of a mitigation install are set by the local building department, so confirm with the jurisdiction the property sits in. Where the finishing scope already requires an electrical permit, adding the mitigation fan circuit to that same application is simpler than pulling a second one after the walls are closed.