No actual circuit provided??
Most likely you need to add a fairly low value pull-up or pull-down resistor between the switch input at the arduino and its power or ground (opposite to the other side of the switch). Try 1K or 470 Ohms.
If that alone does not work, add a 0.1uF capacitor in series with a ten ohm resistor, across the switch input at the arduino.
More effective filters are possible, but we need the exact wiring details to be able to make better suggestions.