Helsingør can be reached from Copenhagen International Airport (CPH) at Kastrup by a commuter train in about one hour. Copenhagen Airport is the main hub of Scandinavian Airlines (SAS, a member of Star Alliance) and Norwegian Air Shuttle in Denmark. From other places in Europe, you may also go to Copenhagen or Helsingør by a train, car or boat. From Helsinborg in Sweden, the ferry boats will bring you to Helsingør in about 20 minutes. Note that if you wish to go from Helsingør to Helsingborg or from Denmark to Sweden in general, a valid passport or ID card is required since last year even for Schengen citizens.
You can get to Helsingør by one-hour train trip from the International Airport of Copenhagen (Kastrup Airport; CPH Lufthavn) to Helsingør St. During day time, trains go from Copenhagen CPH airport to Helsingør several times per hour. Some are direct and others are with a transfer at Copenhagen central station (København H). You need to have a valid ticket before entering the train. Travelling without a valid ticket will result in a fine of up to DKK 1000.
Only the bus tickets can be purchased by cash in the bus. No change returned, however. For trains and metro, tickets can be bought from tickets shops, ticket machines and online. Tickets can be bought using the ticket machines close to the train terminal. There is also DSB (Danske Statsbaner; Danish State Railways) ticket office in the arrival hall. You can also purchase a ticket in advance by using the link , but you should then notice that the ticket is only valid for the specific time chosen.
Detailed description of how to reach Helsingør can be downloaded here.