There are thousands of aircraft out there with transponders but the owners have decided to not equip with ADS B out. Can they still use the transponder after the deadline or is it essentially a boat anchor?
Yes, they can still use their transponder (as long as it still has a current 91.413 check), but they'll have to stay out of all the ADS-B-out-required airspace including all Class B, the Mode C veils around Class B, and all Class C.
Ron is correct. I will also add that flights at and above 10,000 feet MSL in class E cannot be planned without ads-b out installed. Otherwise, the transponder requirement regulation, 91.215, is still in place and remains unchanged. If pilots of non-ads-b out equipped aircraft need to enter ads-b airspace, FAA has implemented a program called ADAPT. Pilots may request an authorization to deviate from the rule online here - https://www.faa.gov/nextgen/equipadsb/adapt/