It's the front brake light switch. 
That said you're issue i.e. bike cutting out when the clutch is pulled, could well be related to the clutch switch which is the same only on the left hand side below the horn button. Are you sure you've not got the starter and horn buttons mixed up? As you question states starter button and that would make the switch pictured the brake switch. The brake switch failing would just mean no brake light when you use the front brake, it would have nothing to do with the cutting out when the clutch is pulled.
But if it's the clutch switch that's pictured it would, as it's linked to the side stand and gear position switches. These prevent you starting the bike with it in gear and/or riding away with the side stand down. When either one of these starts failing they can intermittently cut the bike out when you pull the clutch when changing gear.
That said you're issue i.e. bike cutting out when the clutch is pulled, could well be related to the clutch switch which is the same only on the left hand side below the horn button. Are you sure you've not got the starter and horn buttons mixed up? As you question states starter button and that would make the switch pictured the brake switch. The brake switch failing would just mean no brake light when you use the front brake, it would have nothing to do with the cutting out when the clutch is pulled.
But if it's the clutch switch that's pictured it would, as it's linked to the side stand and gear position switches. These prevent you starting the bike with it in gear and/or riding away with the side stand down. When either one of these starts failing they can intermittently cut the bike out when you pull the clutch when changing gear.
Later