Well, my first suggestion is to clean the carbs. You could try running SeaFoam in your gas for a few tanks in hops it will clear it up, or just pull them and clean them. It sounds like a pilot jet is clogged. Also, you'll want to sync the carbs after you verify they're in good clean/working order.
You could also have weak ignition. I suggest a new AGM battery (if you have not done so already) and switching to Iridium spark plugs (part # in my sig). Even if these don't fix your problem, they really are good for the bike.
Other than that, let us know what you've tried to resolve the issue. There are a battery of resistance tests/etc you can perform on the ignition coils, pickup coils, ignition cables, etc.
1986 Kawasaki Vulcan 750
NGK Iridium Plugs #7803/DPR7EIX-9
Duralast Gold ETX15 AGM Battery
Coastered & Shaved
Metzeler ME880 [110/90-19, 170/80-15]
Balance Dampers Replaced
Rebuilt Forks w/ Progressive Springs
VN750.com Grill Cover
Emgo 23-92411 Handlebars
MOSFET FH012AA R/R
1990 Kawasaki Vulcan 750
1998 Honda VFR800 FI
2014 Honda VFR800F
1989 Pontiac Firebird Formula 350