The ein in eindringen is an in/into. That's true for a lot of verbs:
dringen (to pass a gap) — eindringen (to enter by passing a gap)
fahren (to drive) – einfahren (to enter by driving)
spritzen (to squirt) – einspritzen (to inject)
Sometimes the ein is an intensifier:
lesen (to read) — einlesen (to read until you understand a topic)
reden (to talk) — einreden (to convince someone about the untruth, to talk against an order)
stellen (to put) – einstellen (to adjust)
Sometimes the derived meanings are hard to guess:
fallen (to fall) – einfallen (to come to mind, to invade)
gehen (to go) — eingehen (to incur, to gamble, to die – plants, sometimes animals)
laufen (to run) — einlaufen (to run into, to shrink by washing – clothing)
So, to answer your question: Whenever someone or something enters something by passing a gap or leak, use eindringen, if not, dringen.
Note there's also the verb durchdringen (emphasis on first syllable, separateable) which means the same as dringen, but passing the gap against some resistance, and, to confuse you by purpose, durchdringen (emphasis on second syllable, non-separateable) which means to soak, to interfuse.