Can I put a heat pump water heater in my finished basement? Won’t it take away from the efficiency of heating my house, by exhausting cool air into the basement?

Yes, it’s fine to put the heat pump water heater in a finished basement. You should just make sure the cool exhaust air from the water heater does not blow directly into commonly occupied areas. Yes, this will add a small amount of energy to the home’s overall heating load. However, this amount of cool air from the heat pump water heater will have a minimal effect on the home’s overall heating system performance/consumption. In addition, the cool air will help keep the basement cool in the summer months, reducing the energy used for air conditioning.