Basil doesn't like cool weather. The other plants you mentioned are not that sensitive. Are you planting it when the night time temperatures are consistently above 55ºF? Does the nursery you buy it at have the plants inside the greenhouse or do they have it out while it is too cool? Try planting a little later in the season when it is warmer. Here in Northern VA, the time to plant basil is around the 3rd week of May, sometimes a little later.

-F. DeBaggio

Buying basil plants are usually forced grown. This is what many look like at the beginning, but die out and yellow soon after. Leaving it in small pots will cramp its roots, and the plant cant retain much water. Also when forced grown, they use fertilizers, and these are short term. Its much better to plant seeds in small pots indoors about early April, and then transplant outside in June. By the end of July, the plants will probably look very healthy, and cost a lot less than ones you buy. Also, if a branch is broken off, it can sometimes form its own roots. Basil in NH is only a summer plant.