The next International airport is La Romana (LRM). It is about 20km from Bayahibe. There are some charter flights from Europe there and of course "normal" Airlines fly there too.
Alternativly the airport Punta Cana (PUJ) is an option its a bit further away.
At the airport you have to go through the immigration and customs controls. You need to buy a "tourist card" wich costs 10.-- US$. Sometimes you can get the card before you leave your homecountry but you will pay more and won't save much time. It is possible that you don't get the tourist card with your own name on back, but that's no problem when you leave.
From the airport you can get a Taxi to Bayahibe wich will cost about 10 US$ or walk to the main road (ca 100m) and catch a Guagua wich will cost about 20 RD$ but the Guaguas are often crowded and possibly you will have to wait some time before you get one.
The guaguas (a kind of taxi-bus) connect Bayahibe and La Romana (stopping at the airport and anywhere in between on request). They run about every 20minutes through the day, the first at about 8 in the morning the last one at ca 6pm. A bit less frequent they run to Dominicus too.
In Bayahibe they start at the ghagua station in the middle of the village, near the big colmado, in La Romana near the supermarket "Jumbo".
The guaguas are mostly minivans and if you are lucky they aren't very full. But if necessary they put as much people in them as possible wich makes them very crowded and uncomfortable...
If you get near to the station looking like a tourist (wich you will do unless you're dominican) they will want you to use a taxi and pay much more (Bayahibe-La Romana ca US$ 20.--) but just say you want to use the guagua and it will be ok. The fare to La Romana is RD$ 25.-- so very cheap and will be collected before you get off the guagua. In La Romana the driver will point out where you need to stand if you want to catch a connecting guagua or bus to somewhere else, as a bus terminal or something like it doesn't exist.
If you want to go around you can take a taxi or a guagua. To visit the inland, where the roads are dirt, it's better to go by a land rover. We went by this pickup.