There are a host of restaurants lining the Mekong and many are good. The Vieng Kheam Khong is very popular and this is borne out by the fact that you will see many local people there as well as foreigners.
It is also very reasonably priced. There is an area by the roadside but far nicer is a softly lit wooden deck overlooking the river where you can contemplate the twinkling lights of the boats as they glide by silhouetted mountains lit by moonlight on the far shore. The food is authentically Lao and authentically delicious. The Luang Prabang sausage is interesting if you have also tasted Chiang Mai sausage in Thailand. It has an almost woody taste, gently seasoned as compared to its fierier Thai cousin. The Lao stew (whether fish, chicken, or pork) is also a good option. Beware, although it is not that spicy they do chuck in whole unseeded chilies and if you accidentally bite into one then you might find yourself leaping feverishly into the fast-flowing waters of the Mekong to put out the fire. It does work pretty well as a momentarily distressing cure for blocked sinuses or indeed the common cold.