There's a quick cover of key things here.
Pick vs finger: a pick will give you a bitier more aggressive sound, but using your fingers is ok too.
Attack: most punk is played aggressively, and if you use fingers or a pick, be heavy handed. Hot responsive picks don't hurt.
Clank: a lot of punk bassist sound uses clank, aka strings hitting the fretboard. If gives you that percussive sound you hear sometimes. Just lower the action...
EQ: mid heavy, you want it to sound aggressive and mean. Mids do that well
Overdrive/distortion: a lot of punk bass tons are distorted or overdriven. This does two things actually. Number one is the obvious, it sounds overdriven. You can have a light overdrive or a nasty distortion, but find what you like and fits with the band. The second thing this does is compression. Since you're hammering the strings, a some compression can be a good thing. Of course if you have a tube amp you don't need to bother with these mostly, but for an SS amp a good overdrive pedal can help a lot.
Strings: round wounds are almost needed. Personally I love prosteels. They're rich, clear and have kick to them.
Final thing... Do what ever you want, these are just suggestions for the "standard" punk bass tone. If you wanna rock out with flats, fingers and a clean tone, do it.