I'm not sure if you're asking if you need to do lower body ST in general, or if you should do it on days you run.
It is very important you keep strength training your lower body. But, if you run the same day you do upper body, I don't see why you can't do lower body the days you don't run. Especially if you find yourself fatigued either running or ST. And, if you become sore from ST, running is a great way to flush that lactic acid out of your muscles the next day!
I had an injury during my marathon training, and the best guess of what caused it was an inbalance of flexibility and strength in my lower body. So, make sure you stretch well, and do some lateral lunges!!
Patience and Fortitude - Mom
There ain't a thing I've faced that's been too much for me - Classified "Inner Ninja"