Most home WiFi router/access points have this very annoying hold-on-as-long-as-possible behaviour.
My solution was to get a stronger WiFi router placed centrally in the house, which works for regular surfing nearly everywhere in the house. It's still not strong enough to stream HD reliably everywhere in the house though.
For that I put two additional weaker $13 access points far from the central one, in two spots where I might be streaming HD. They are on different SSIDs so I have to manually switch over, for the occasional times I need to stream HD in those spots.
The other solution would be to buy 3 Ubiquiti UniFi access points which have that seamless handoff functionality, but I didn't feel like spending $200+ at this point for 802.11n UniFis. Maybe in a couple of years when 802.11ac is mainstream and the Ubiquiti UniFi 802.11ac models are cheaper, I'll get three of those.
BTW, I too had my router in the basement, and it was a much bigger problem. Now I have my newer WiFi router (with the stronger signal) in the living room, in the middle of the house.
Is it really not feasible to put an access point in your living room? You could run the CAT6 cable outside for example. It might require drilling through the wall in your basement and in your living room though. Another option is to drill a hole through the floor of your living room / ceiling of your basement to run cable. Or else you can just buy the Ubiquiti stuff.
In the meantime you could try reducing the signal strength of your basement access point, and then buying a very strong access point for your home office. That way, in your living room and in your office, you're always just using the office access point. The only times you'd use the basement access point is when you're in the basement.
P.S. I turned off 5 GHz in my house completely. The range just wasn't good enough compared to 2.4 GHz, so in spots further away from the router, 2.4 GHz was consistently faster and more reliable than 5 GHz.