Yes, leave it on the vine until the vine is dead. You can also leave it on the vine all winter to cure out. When the seeds rattle inside, you know they are fully ready.

If still the goud's stem and part of the vine that it is attached to are live then you should leave it on the vine.
But once thevine is dead, the gourd can still be left where it is or picked and brought elsewhere to mature and dry.
The color of bottle gourds will gradually fade away to pale green, at which time molds will start changing the color and staining the gourd in some interesting patterns.
A ripe mature BH gourd will have hard shell and withstand rain, cold, snow freez...
When the seeds inside start rattling, the gourd is albost dry and you can work on it or leave it for few more months.
If you do not want your gourds skin stained by molds, you can wipes then with chlorine soulution, to kill the molds. I personally like the work, foot print snd signature of the molds, in a more controlled way.

how can you tell the seeds are good?
Easy, if the gourd was fully matured, the seeds will be good. The simple way to know for sure is to examine the seeds. Do they look fat and heavy?
Crack open some seeds. Do the have good size kernle?
If you got "Yes" for answer, then the seeds are fine.