This is a very old question, but I thought it might be good to answer anyway. When you said curved lines, I assumed you meant drawing a great circle. There is an example of doing exactly that in the basemap documentation, which I have modified to make a little more easy to modify yourself:
from mpl_toolkits.basemap import Basemap
import numpy as np
import matplotlib.pyplot as plt
m = Basemap(projection='cyl')
p0_ll = -73.98, 40.78
p1_ll = 0.08, 51.53
m.drawgreatcircle(p0_ll[0], p0_ll[1], p1_ll[0], p1_ll[1],
linewidth=2, color='b')
m.drawcoastlines()
m.fillcontinents()
plt.show()

Note that the great circle method cannot handle the crossing of the edges of the map (as mentioned in the documentation), which, although clearly documented, is a pretty major flaw IMHO.
Hope that helps somebody,