From Bavaro to Santo Domingo, there are express buses running at 07h00, 10h00, 14h00 and 16h00. It is an excellent service and when last I used it (3 weeks ago) I believe I paid RD$250 one way.It is about 3.5 hours and they stop to buy refreshments in La Romana for about 10 minutes. They leave from the main bus station in Bavaro. As far as safety goes, there is never a guarantee in life, but I have travelled alone on the buses many times and have never had a problem...more weirdos on the TTC!
For a bit less (approx. RD$160), you can change buses in Higuey, but you need to allow extra time. I would suggest the express service having done both!
Hope this helps! Enjoy!