Noteworthy places to visit are the castle of Melfi which of course is in the town of Melfi and the archeological museum in Potenza which has an interesting section of Greek and Latin inscriptions. Matera Is an interesting town for two reasons one is, it is where the Mel Gibson film passion of the Christ was filmed and also because the Sassi district is a unesco world heritage site where you can see homes which were carved into the hillside by the residents of long ago. Like all parts of Italy the Basilicata region has its share of Roman ruins to see, more so near the bigger towns then in the more remote areas. The easiest way to find them is to do a search of the town you are interested in visiting and you will find a lot of information for that town including information about any ruins that may be in the area.
It would be great if I could provide information and a link for every city and every museum in each region but not many if any websites could really do that. With that in mind if you want information on any place you find interesting you could try wikipedia which has thousands of people from all over the world providing them with information. Also don't forget to come back to do your hotel search and booking once you have picked out the city you want to visit or just bookmark the links page.
Although the region isn't heavily populated it doesn't mean none of the cities or towns have museums because they do. There aren't as many as other regions and I listed the few museums in the region with the town they are in.
Hopefully the information provided will help you to better decide where in Italy you may want to go and by using the Planning your Italian vacation page will make planning your Italian Vacation a breeze.