Check the switch in the lever, you'll see it, it's where the two wires connect, on the underside of the lever.

What year?

I had this problem when I first got my 87.  This may or may not help, but this is what I had.

Inside the switch is a very little contact, number 8 in this diagram:

ihttp://www.bikebandit.com/1987-suzuki-ls650ph-savage-p/o/m6323#sch239721

It is very small and will jump out when you are not looking.  In fact, it ran away from the PO on my bike.  I tried a local shop and he couldn't or wouldn't get the part for me.  I am sure you can order one from here.

Then make sure the tiny wires on the contact between parts 8 & 9 are OK.  They about as big around as a hair or two and will break with little resistance.  If you put a big blob of solder on there you will blow fuse after fuse, trust me on that one.

I had to stand on my head, hold the wires with one hand, the plate with one hand and the soldering iron with one hand.  It was quite a feat, but I finally got it.

And I am sure you can too.

Of course, this might not be your problem.