This type of problem is most commonly caused by a loose or defective stem packing washer.
If you will examine the attached illustration you will see a side view of a typical tub/shower mixer unit. For a two handle mixing unit both the Hot & Cold handles will have a "Stem" as illustrated. For a three handle unit there is a third stem in the center that serves as the diverter to direct the water either to the shower head or the tub faucett.
In the illustration you will note where the control handle shaft enters the outer end of the stem there is a "stem packing nut". Directly under that nut there is a fiberous packing washer that surrounds the shaft and prevents water from leaking out around the shaft. Normally when a leak such as you describe occurs you can stop the leak by tightening the "Stem Packing Nut" which will in turn compress the internal packing tighter against the stem shaft. Typically when the stem packing is loose the water will leak around the stem shaft and drip down inside the "trim escutcheon" where it then flows forward toward the handle, leaks inside the handle, then leaks out from the back of the handle down to the tub or on the tub apron.
To correct the problem you must first remove the handle. Typically the center of the handle will have a small plastic or metal button that has the H or C index printed on it. Those buttons are usually just press fit into the handle and can easily be pried out. Under that button you will find a screw through the center of the handle that locks the handle on the end of the "Stem Shaft". Remove the screw, then tap the handle lightly to loosen any scale that might be present and pull the handle straight out. You will probably need to wiggle it as you pull forward to get the handle off. In some cases if the handles have been in place for a long time they are very difficult to remove and you may need to use a "Handle puller" to remove the handle. (The handle puller shown in the illustration can be purchased at any local hardware store for about $8)
Once the handle has been removed you may be able to access the stem packing nut through the end of the trim escutcheon with a plumbers socket. (The set of plumbers sockets shown in the illustration can be purchased at any hardware for about $10 for the complete set).
If you do not have the plumbers socket you can remove the "Trim Escutcheon" then tighten the packing nut with an adjustable wrench. (some trim escutcheons are screwed on and some are just a slip fit. Either way, to remove, try turning it as if you were unscrewing it, (If it is a slip on type it will come off quickly, otherwise you continue screwing it off.)
Once you have access to the stem, tighten the stem packing nut about 1/4 turn, then turn the water on and watch where the shaft enters the stem. If it is still leaking tighten it a bit more until the leak stops.
Once you have the leak corrected you can replace the trim escutcheon and the handle and your done.
Both the plumbing sockets and the handle puller are under $20 and are a very handy tools to keep in any DIY tool box.