Here's somewhat of a how-to/check list...
As with any quad;
Frame
1) Check the frame thoroughly for bends or repair welds that don't belong. On the Banshee, particularly around the rear seat mount where the pipes mount to the frame and around the foot pegs.
Put it up on a box or crate to get the wheels off the ground then check:
2) pivot bearing for wear (swing arm movement side to side.)
3) carrier bearings for wear (axle tilt side to side)
4) sprocket/brake hubs tight on the splines (no rotation forward or back relative to the axle)
5) Ball joints and a-arm bushings on the front - no free play
Engine
6) Compression check - dependent on altitude. But the difference between cylinders shouldn't be more than a pound or two. Check the table at the bottom of this page: NosMachine Banshee Heads
Granted these are #s. for fresh bores so it's a point of reference only.
7) Check intake boots between carbs and engine for signs of drying/cracking
8) Starts and runs smoothly through all 6 gears. Banshees are typically difficult to get into neutral while the engine is running. I find it easier on mine to do it from 2nd gear.
9) Engine case whack around the front sprocket
10) if there's a skid plate, you'll wanna see if the engine case underneath it's ever been cracked and JB Welded or anything.
11) lights and electrical; look at the harness to see if it's frayed or hacked up.
If anyone can think of anything i forgot to add, by all means chime in.
Hondapol's point about looking at the overall condition is a good one too. If it's been well maintained in general, that's always a good sign.
'01 Sportsman 500 HO
Project - 2" lift, 1" spacers
'97 YFZ350J Banshee
Pro Circuit Platinum Exhaust
Open Airbox w/K&N filter & Proflo Adapter
VForce 3 Reeds (Awesome!)
Stock Carbs with #30 pilots, #360 Main Jet, & Vito's needles, clips pos. 3/6 ~900 ft above sea level
Wiseco 65mm Pistons
Hot Rod Crank
Cool Head w/21cc Domes