Not sure, but my assessor stopped by soon after my basement passed final inspection. He had his clipboard and tape measure in hand. He didn't actually come inside though. He asked me how much finished sq footage. Whether it included a bath or not, what kind of flooring and walls, fireplace, etc.
A couple weeks later I received my new property value assessment.
My new taxes included a new porch as well, so I'm not sure how much the basement impacted it.
You probably could ask your neighbors just to get an idea for your area.

This must be a regional thing. In this area it doesn't mean a thing as far as tax assessment, I guess because they don't count that square footage. It does help with appraisals though.

It will depend entirely upon your local tax structuring.

In most areas any work that requires a permit can effect you assessment. Once they do the final inspection for your building permit it flags the assessors office to re-assess the property.

In my city, if you look at the online records for recent building permits...it lists the added value. I think finished basements in 2300 square foot colonials...which ttypically only amount to maybe 1000 square fot finished...were adding $15,000. These were homes in the $340,000 range.
I think I may have neglected to get a permit for mine...I did all the work at night and was taking Ambien and don't remember any of it.

Blaming it on the Ambien? Good one.

Be careful: I had an terrible experience. I purchased a home advertised with a full finished basement. Much to my surprise I received a letter shortly after from the county. They had pulled real-estate listings compared descriptions and sales data. The previous owner had finished the basement but not permitted it. It almost doubled the sq ft. Therefor the same thing happened to the tax bill. Over a 45% increase in property taxes. My fault for not checking before I bought it. You can bet my first stop next house is the tax office.

Where I live (Long Island, NY) you need a permit for EVERYTHING - even remodeling an existing bathroom. That being said, NO ONE gets one for that or for finishing an existing basement.

BUT, when you go to sell the realtors are REQUIRED to report any inconsistencies between the what exists in the house at the time of listing and what the Tax Assessor's office has listed. The catch is that the seller gets fined, the new owner pays more in taxes. BUT, as far as "finished" basements go, you MAY be able to fight it because a DIY "finished" basement is not necessarily "liveable" - that requires PROFESSIONAL waterproofing inside and out (and permits for drains etc).

My former neighbors next door supposedly finished their basement, but I don't think it was included in the listing or on the new guy's taxes. I live too close to the water with a high water table for anyone to have a truly "liveable" basement.

I can't see how a truly finished basement anywhere would double the tax assessment, since ANY underground space, no matter how well-designed, will always have a greater risk of destruction than the above-ground portion. Also, a house with an unfinished basement is already taxed higher than a house on a crawlspace or slab, most likely because they assume people will "finish" it in someway.