There are two ski resorts in Belianske Tatry - Strednica (www.strednica.sk) with 7 ski lifts and slopes mostly suitable for average skiers and beginners, very good for kids. Ski and snowboard school on site. The snow usually stays here quite long from December till March, mid April. The other one is Bachledova dolina with slopes also for more demaning skiers.
In the viallge Ždiar you can go on a ride on a horse-drawn carriage (summer) or sledge (winter). The whole ride lasts about 60 min and takes you through the village and under the mountains. A very romantic experience. If you're staying in hotel Kolowrat they'll book it for you.
Get a map of Belianske Tatry, there are hiking tours from small ones (about an hour) to long half-day or whole-day trips. Good outfit and shoes are a must as well as packing something warm and waterproof even in winter, the weather can change here dramatically. One of the trous I'd recommend is from Tatranska Koltina to the mountain cabin Plesnivec. For a whole day tour there's a nice hike from Tatranska Kotlina to Tatranska Javorina.
If you don't have a car the only way to get to Belianske Tatry is by bus from Poprad. There are no trains here. For information on time tables check www.cp.sk where you ahve to write the place of departure (Poprad) and place of arrival (Tatranská Kotlina, Ždiar or Tatranská Javorina). To get to Poprad, either by plane or train. Easy connections from Bratislava.