Should I finish my basement

...and you have a right not to let unwelcomed to your dwelling anywhere you live in US. I used to live in NY,NJ as well and dealt with those who have right to come in by simply not letting them in. Entering the dwelling without permission of owner is called trespassing weather one has or doesn't have a right to walk in. Technically I have a right to use weapon to protect my dwelling (2nd ammendment) and if it's not firearm, than 2" adjustable wrench or cro-bar may also work.

Untechnically ,in most states you try any of that with assesors you will end up in jail.
Some will pull eminent domain on you which the current Supreme Court has ruled they can do for ANY reason whatsoever.

Schubert, if that is to say, I can impersonate assessor, enter the property and at gun point take every valuable item, money jewelry (maybe wome precious amps turntables whateverwhatever) and drive away. How should I technically know that I'm dealing with assessor, Social Services and not burglar??
I didn't let in Social Services not because the reason of visit didn't make sense, but simply because I didn't expect them and I didn't have any written note about scheduled visit!
When I shop for property I'd always look for lower value property in order to modify it and improve the way I like to live and it doesn't mean that I will let anyone enter without prior authorization and introduction. Scheduled visits with me is REAL PAIN in anus! I'm either on vacation, or out of town, giggin', DJ-in', working, running late, sick or simply not in the mood to see anyone...
In most of State laws they would state that an assessor HAS RIGHT to enter private property, BUT

The assessor or an employee of the assessor shall not enter the interior of any structure on any real property as part of the inspection to assess such property without permission

. My permission they usually wait FOREVER if not longer. When conversation at door turns in to 2nd ammendment business, weather or not one has or hasn't right to enter, cognitive individual would make quick U-turn and get back to other activities or look for easier prey.

If you finish it you will turn it into an echo chamber. It probably sounds better the way it is with exposed framing. The worst sounding rooms I've ever had were sounds rooms I purpose built and they turn into an echo box and giant bass collection bin which is expensive and hard to figure out how to treat. The best sounding room I have is my workshop which has tons of stuff in it which gives lots of natural diffusion which is a more pleasing sound if you ask me

Don't forget about dust issues associated with unfinished basements.