Easy and reasonably priced:
Boca chica airport transfers, dominican republic, punta cana airport transfers
There is some security issue with the cruise ship port and a fax needs to be sent to them in the days before your pick up with names and cruise ship info to be given permission to access the area, but the pros are familiar with the process and can arrange it with no trouble to you. It will be easy for you to book a pick up from the port with a return to the LR airport.
You could catch a cab, they certainly are waiting, but though there is a standard rate for this trip with the syndicate (about $90) a friend of mine was charged $120 since the cab driver had to return to his home base without a fare (that was the other way around--from the Punta Cana airport to La Romana, but it could still happen if the cabbie wanted to assert a position). To pre-book, you know in advance what the cost will be and you can decide if you're prepared to pay it or not.
For the 90+ minute drive, the cost is about $95 each way for up to 5 people. I've used them since 2006 for transfers from Santo Domingo and Punta Cana to La Romana.
Kaki