If you paid $370,000 that is your cost basis. If the property had declined in value when you placed it in service as a rental, you have the option to use either your cost basis or the current value when placed in service. Tax assessments are not a good way to determine value, as they usually are not accurate. A better determination of value would be to look at comp sales made around that time.
What the property tax assessment is good for is to allocate your cost basis between land and improvements. So, if you used the ratio of your assessment to allocate your $370,000 cost basis as $92,500 for land and $277,500 for the improvements (building).
