Try running the starter by itself, use jump starter if you have one otherwise a set of jumper cables to a vehicle, put the ground to the frame and then touch the positive to the terminal on the starter, does it spin over? If so then its probably a bad solenoid, which you can test further also. The solenoids usually go first on these, but the starters will burn out brushes and become unreliable as well. The solenoid is an easy fix, and cheap. The starter you can get a rebuild kit for or just replace for about $40.
But definitely figure out which is the first. Whats the noise its making? Do you hear the solenoid click? Sounds like the battery is done for. Replace it with a good AGM battery! I went through two cheap $40 lead acid flooded cell motorcycle type batteries before i bought my AGM, cost more but well worth it. Check out the ETX15 Big Crank, that fits the 200E as well as my 200ES. Its made a huge difference in cranking power for me.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....