Hi, Paul and the usual suspects are:
1. Severely discharged or a damaged battery should have 12.5 volts or more and be able to pass a proper "LOAD" test if necessary, you may have a preliminary reading of 12.5 volts or more but little or zero amperage the battery is faulty and must be replaced, AGM batteries fail in this scenario more so than lead-acid batteries.
2. Failed alternator/generator and or voltage regulator.
3. Loose or corroded battery terminals and or cables especially the "NEGATIVE" cable, look for loose, corroded, or broken connectors inside the cable harness at "BOTH" ends.
4. Failed main circuit breaker or ignition switch, check for loose connections and continuity.
5. Failed system and or ignition relay, check for continuity.
6. Failed ignition coil, stator, magneto, ignition/electronic module.
7. Failed CKP, CPS, CMP, MAP, TPS, or BAS sensor, corroded, loose or broken wire connector pins/sockets.
8. Throttle cables and or idle speed improperly adjusted hot idle speed should be 950 RPM to 1000 RPM.
9. Faulty neutral, side stand or clutch lever safety switch.
10. Faulty or corroded kill switch.
11. Accelerator pump damaged or not working.
12. Water or dirt in the fuel system, carburetor or filter.
13. Restricted, blocked or kinked fuel line.
14. Fuel tank empty.
15. The gas cap is not venting properly or fuel tank venting system blocked, loosen gas cap and go for a test ride.
16. A failed fuel pump, pressure regulator and or fuel injectors.
17. Vacuum line from intake manifold to petcock broken, cracked, or not attached, carburetor vent line plugged.
18. Needle and seat stuck closed in the float bowl.
19. Petcock clogged or damaged.
20. Lean angle switch is faulty or needs adjustment.
21. Catastrophic engine failure, perform a compression test.
For more information about your issue and valuable "FREE" downloads that you will need for viewing or printing please click on the blue links below. Good luck and have a wonderful day.
Suzuki DR350 Service Manual