drawmeridiansは、経線（子午線）とそれに対応する軸ラベル（緯度の数値）を描く。
線無しで、軸ラベルのみを書くためには
map = Basemap(projection='cyl',llcrnrlat=-90.0,urcrnrlat=90, \
llcrnrlon=0.,urcrnrlon=360.1,resolution='c') #Equidistant CYLindrical Projection
map.drawmeridians(np.arange(0.,361.,120.),labels=[0,0,0,1],fontsize=10,linewidth=0.0)
軸ラベル無しで、線だけ描くには
map.drawmeridians(np.arange(0.,361.,120.),labels=[0,0,0,1],fontsize=10,linewidth=0.0)
緯線を描く関数は，drawparallels．
drawmeridians draws the meridian and the corresponding axis label (numerical value of latitude).
You can draw only the axis labels, without lines by
map = Basemap(projection='cyl',llcrnrlat=-90.0,urcrnrlat=90, \
llcrnrlon=0.,urcrnrlon=360.1,resolution='c') #Equidistant CYLindrical Projection
map.drawmeridians(np.arange(0.,361.,120.),labels=[0,0,0,1],fontsize=10,linewidth=0.0)
You can draw only lines, without the axis labels by
map.drawmeridians(np.arange(0.,361.,120.),fontsize=10)
"drawparallels" function draws the meridians.