This is not a recommended option and having anything more than hand luggage would be a problem. There are buses from Kolkata to Krishnanagar. From there you would need to take a different bus to "Navadvipa Rail Gate". From there you would need to take a rickshaw. Buses depart from the airport "Ek Nambar Gate", an Rs 10 rickshaw ride from the terminal. The buses are normally going to Malda or Borampur with a stop in Krishnanagar.
There is also a bus from the ISKCON temple in Kolkata to the ISKCON Mayapura temple (the opposite side of the river), but as the Kolkata ISKCON management is extremely offensive to Srila Gurudeva I would sooner walk to Navadvipa instead.