My toilet handle has a trickle of water when it fills up but the rest of the time it is dry.What can I do to correct the problem?

Your tank has an overflow tube which should stop the water from filling this high. You can adjust your fill valve to stop filling 1" below the top of the over flow. This may help you understand the working and the fix o many toilet tank issues. http://www.youtube.com/watch?v=08mvV3pDxtE

Ballangerb said it. The overflow tube is too tall and your float is also adjusted for higher-then-needed water level in the tank. Adjust the float, cut down the overflow tube about 1" bellow the holes in the tank.

Also, you may have water spraying onto the handle inside the tank from fill valve. Take lid off and observe fill. See if there is water spraying sideways during the fill.

What do you do if there is over spray when the toilet fills?

The water should be exiting out a tube down near the bottom of your tank. Where do you see over spray coming from, possibly the small refill tube that goes into the over flow tube? A Fluidmaster 400A is about the most common and best fill valve. If you have this the link will help you repair it. http://www.toiletology.com/troubles.shtml

What do you do if there is over spray when the toilet fills?

Water spraying from the fill valve indicates the fill valve is broken and most-likely has to be replaced.

Also, the leak may be from the small 1/4" fill tube that delivers water to the bowl during fill up. This tube should be attached to the overflow tube. This small tube sometimes slips loose from its mounting hook shooting water all over the tank. Make sure that 1/4" tube is properly attached and inserted into the overflow tube.

First tell me what type of ballcock you have,(see image) Next, open the lid, give it a flush and tell us where it's leaking from. Regards, tom