First of all, a bit of rattling is absolutely normal and I always have the impression that my toms don't sound right if the snare remains silent.
There are a few things you can do to reduce the rattling if it's too much...
- Try to avoid direct sound from guitar or bass amps, i.e. don't place them in the snare's direction. On professional shows you sometimes see some plexiglass shields preventing too much direct sound from other instruments or monitor speakers. Those are meant to help the microphone isolation and prevent the resonance problem.
- Play around with the overall pitch of your snare, trying to avoid the direct resonances with the rest of your kit or your band colleagues.
- It can help to adjust the resonance head of the snare to be somewhat out of tune. Usually (especially if your snare has a "snare bed" for the springs), the four screws next to the springs are tuned to have lower tension than the rest of the bottom screws. This reduces the sensitivity of the springs a bit.
It all depends on your playing style and the type of music. In classical music, it's quite common to have the snare very sensitive as you play very softly. This means that you need to turn off the snares as often as possible. In rock music on the hand, you usually hit harder and don't have so many fine embellishments, so you can tune the snare to be less responsive and rattling.