From Punta Cana airport walk up to the main street along Terminal A (if you don't have money yet there will be an ATM). You will see signs stating Autobus. Eventually ending up at a big roundabout from which you can walk to your right. You will find a small bus station there. Here you can jump in the next mini bus to Veron de Cruz. Expect to pay around 60 Pesos. When you arrived in Veron de Cruz walk right again. Behind the police station is another bus station where you can grab a bigger bus to Bavaro for around 40 Pesos. Tell the driver where to stop. To be sure pick a bigger resort to make clear where you want to go.
Most hostel are located in the same area so it can make sense to check the locations beforehand.
Bavaro Hostel Dorms from 12$ or 550 Pesos. One fan per room so really hot at night especially the upper bunks. Smelly shower. But seems most backpackers go to this place. Close to the beach.
Punta Cana Hostel AC Dorms from 15$ or 790 Pesos. Close to the beach.