The only solution is to have a rigid, massive dartboard (I mean lots of Mass and not 10m across!) which is decoupled from the surrounding by a 'lossy', bungy medium. Walls and floors transmit sound very well so it must not even rest directly on a hard chair. To reduce sound traveling through the air you could use heavy furnishings around the board, like a cubicle or large diameter tube with sides lined with carpet, blanket etc. etc.
I guess you could try for some sort of sound cancelling system, with a microphone placed right on the board (to detect the vibrations and amplifier / loudspeaker right behind it. This would need careful layout to avoid howl-round, of course, but if you are electronically inclined it could be fun.
I think the essential thing is that you spend so much time on this project that it keeps you away from the actual dart playing. That would keep your neighbours happy.
Failing that, you could take up Grand Theft Auto instead of darts.
You could, perhaps, challenge your neighbours as to how loud you are actually being. Get of them to throw darts at your board whilst you are in their flat, listening. Merely making enough noise to be 'detected' doesn't actually constitute a 'nuisance' for a reasonable person. But that's all a social matter and not Physics.
I notice they are downstairs - so a thick insulating mat could help.