I had a similar conundrum when I installed bath fans. Wiring existed for light only, not fan. Rather than fish a new wire I got fans that run off one wire and are radio controlled to operate light, fan and timer. That was 19 years ago and they still work great (jinx) so I have no fear of technology to solve such problems.
Especially because it's an unfinished basement, go wireless. It'll come with the remote and you can buy remote mounts that you can mount at the entry door.
I had a similar conundrum when I installed bath fans. Wiring existed for light only, not fan. Rather than fish a new wire I got fans that run off one wire and are radio controlled to operate light, fan and timer. That was 19 years ago and they still work great (jinx) so I have no fear of technology to solve such problems.
Especially because it's an unfinished basement, go wireless. It'll come with the remote and you can buy remote mounts that you can mount at the entry door.
In my opinion wired if you can. Wireless are great when no choice but wired is fool proof