Carol, do you have your basil in potting soil or sand?
They may be too moist? try mixing potting soil 1/2 & 1/2 with sand (or vermiculite) for better drainage...
and it does like periods of shade too...
We grow it on our deck on LI, which gets morning sun, and mid-afternoon shade...
If it is root bound, re-pot it... easy to tell...
just tap pot, removve and if the outside of the soil is full of roots, re-pot!

if it is root bound, remember to score the old roots with a knife, top to bottom, in 4 places...
That will promote new root growth, and force the new roots to grow into the new soil...