Active DI suits for both passive and active pickups, while passive DI suits fine for active pickups. But in some cases, depending on the DI's input impedance (lower = worse), passive pickups can sound a bit muffled because they lose high frequencies. But a high quality passive DI will often sound better than most active DIs. Another thing to remember is how long distances you need to pass. If you need long distances (live shows mostly), then you should use an active DI so as to not lose fidelity. And also, passive DIs are a better choice for keyboards and such.
So, to sum it up:
Active DI
+ Long distances
+ Better "compatibility" in general for both active and passive pickups
Passive DI
+ High quality ones sound better than active
+ Better for keyboards and such electronic devices
- Those with low input impedance will lose high frequencies with passive pickup guitars/bass