You can arrange your transfer to and from Kranjska Gora with the private company GoOpti. It can be also used for arrivals/departures from other destinations such as Klagenfurt airport, Trieste airport, Venice airports, etc.
Another private company offering shuttles from nearby airports is Kofler Sport. The price for the transfer to or from Ljubljana Jože Pučnik Airport is around 80€ per shuttle, which can carry at most 8 passengers. To book the transfer, please contact them at firstname.lastname@example.org.
From Ljubljana Bus Station you can take a bus directly to Kranjska Gora (see timetable and price here), while from Ljubljana Jože Pučnik Airport there are no direct lines to Kranjska Gora. Instead you can first take a bus to the nearest city - Kranj - and then take another bus to Kranjska Gora (find a route here).
Arriving by train either from the south (Ljubljana) or from the north (Austria), the nearest train station is in Jesenice (20km from Kranjska Gora). From there you can take a bus (see timetable here, for "From" select "Jesenice ŽP") or a taxi to Kranjska Gora. You can find the timetable and prices for Slovenian railroads here.