I actually don't mind a basement playroom as long as it's nice and open and there is a lot of light. From what you describe, yours sounds fine. I'm not too sure why so many nannies are against it? In my last position, they had the whole left side of their basement as the playroom. That was where 90% of the toys were so we spent a lot of time down there. There was also a walkout so we were able to walk outside when we wanted too. To make it easier for me, the parent's bought a mini fridge that we kept down there fully stocked with juice boxes, milk boxes, yogurt tubes, etc. I also kept a small basket of snacks for the kids (gold fish, fruit snacks, etc.). I also had a whole changing area with diapers, wipes, powder, etc. as well as a couple of changes of clothes in case the baby spit up, had a diaper leak, etc. I also had the babies swing, pack & play and bouncer down there too.
As long as it's bright and open and not freezing down there, I don't see any reason why you can't ask your nanny if she could spend more time down there.