You have to use upstrokes to play 16ths unless you're at a very slow tempo for punk. Down strokes only on 16ths is like 32nds alternate picking. at 120 bpm (most punk is at least that fast) 32nds are 8 notes per second!! most beginners struggle to get 16ths at 120, which is 4/sec. if you get to more than about 15/sec. it sounds like a blur.
But there’s not really a specific technique u could use, just practice doing what i already suggested and go for it.
Good luck! :)
Vince Rudd.