taxi - about 200 PLN, maybe less during the daytime.
alternative ways (I never used the service of the companies mentioned below, so I have no idea how true all the information is):
http://www.katowice-airport.com/eng/index.html?page=dojazdy,1(AWOS and CarPolonia offer even direct transfers with shuttlefrom the airport to Zakopane, but there must be more passengers, otherwise it's too expensive, for example
Pyrzowce Zakopane
400.00 PLN za samochod (for a car) = do 8 osob (up to 8 persons)
http://www.katowice-airport.com/eng/index.html?page=dojazdy,4(there's a bus to the city center, and Matuszek offers transfer to Cracow, which is probably on your way to Zakopane anyway)
by train (if you somehow get from the airport to the train station):
http://nowy-rozklad.tktelekom.pl/bin/query.exe/en? (in english)
useful abbreviations:
PKP = Polskie Koleje Państwowe (Polish Railways)
PKS = buses (previously a nation-wide state-owned company, now probably privite and divided into regional branches)