Sounds like a bad O-Ring on the stopper.
Here is a good picture for reference.
To remove the stopper, you'll have to remove the retaining nut at the back of the drain body. Once that's removed you'll pull the pivot rod out, this will allow you to remove the stopper.
Now that you have the stopper out, visually inspect the rubber gasket around the bottom of the plug (just under the chrome part). If the gasket is worn or cracked, you'll have to either replace it, or the entire stopper.
If you take the stopper with you to the hardware store, somebody there should be able to help you find what you need.
To put it all back together, put the stopper in the drain, insert the pivot rod (so it interacts with the stopper), then tighten the retaining nut back down. You can make minor adjustments by loosening the set screw that holds the lift rod to the clevis, and adjusting how deep the lift rod goes into the clevis.
WARNING:
Don't over tighten the retaining nut, or you could find that you won't be able to move the lift rod.
1) Use a plastic ziptie (or a short length of insulated 14 gauge electrical wire):
a) Drive one screw (or a nail) part way into the top of the 2x4 blocking somewhere near the pipe. EDIT: I see in the photo that there is already at least one nail in place near the pipe ... just ziptie to that.
b) Then ziptie the pipe to the screw. Zipties are also called cable ties, or tie wraps.
... or ...
2) Use a short length of perforated plumber's tape:
a) Cut a 5"-7" length of plumber's tape. Needs to be the metal variety; thin gauge ok and preferred.
b) Then wrap it around the the pipe
c) Then twist the ends 90 degrees such that they are horizontal and such that they overlap and such that a pair of the large holes align.
d) Then drive a screw or a nail through the aligned holes and into the 2x4 blocking
... or ...
3) What you really want is for the plumbing to be secured at or near the faucet body. To do this, install new '2x4' blocking made out of several pieces of 1x4:
a) Cut two 6" and one 12" lengths of 1x4 (or similar)
b) On one end of each 6" piece, hammer in three 6d finish nails until there is about 1/2" still sticking out. Then use a pair of side-cutters to cut off the six nail heads, leaving 1/8" to 1/4" of the nail still protruding out (the side-cutters will leave sharp ends on the nails).
c) Next, maneuver one of the 6" pieces through the hole then behind the plumbing, into a horizontal position such that the end with the nails is pressed against one of the wall studs and such that the 1x4 is flat against the back drywall.
d) Next drive the cut nails into the stud as best you can by tapping on the clean end with a tool. Do the same with the other 6" piece of 1x4 and the opposite wall stud. You may want to secure them in place by lightly duct-taping them to the back dry wall.
e) Next, maneuver the 12" piece of 1x4 into the hole, then behind the plumbing, into a horizontal position such that it is flat against the two 6" pieces and spanning across them.
f) Next, screw the three pieces together with a half dozen 1-1/4" drywall screws.
g) At this point, it should 'look like' the original 2x4 blocking, but up at the level of the faucet body and with excellent access to secure the plumbing with conventional pipe straps.
h) This method comes with no shortage if 'ifs, ands, and buts'. You'll certainly need to take the basic idea that I've laid out and adjust it to your specifics.
... or ...
4) As an alternative to the 2x4 blocking above:
a) Purchase a largish turnbuckle and a length of threaded rod with matching diameters and thread pitch. The turnbuckle will have two screw-hooks. One will have left-hand threads, the other will have right-hand threads.
b) Remove the right-hand-threaded screw hook from the turnbuckle and discard it
d) Cut the hook/eye off the end of the left-hand-treaded screw hook, then grind a chisel like taper on the cut end.
e) Cut a 7"-10" length of threaded rod, then grind a chisel like taper on the cut end.
f) Screw the square end of the threaded rod into the turn buckle
g) Maneuver the the turnbuckle into the hole, then behind the plumbing the horizontal
h) Back-off the turnbuckle such that the two end pieces extend and dig into the wall studs. Continue backing off on the turnbuckle using open end wrench (or similar) such that the chisel ends dig securely into the wall studs
i) Ziptie the plumbing to the the turn buckle
... or ...
5) Approach numbers 3 and 4 may work if you install the doubled-up 1x4 blocking or the turnbuckle on the front side of the plumbing.
Best Answer
The leak is probably either in the supply lines and valve, or in the drain pipes. It's also possible that a vent through the roof is leaking rainwater. The only way to determine the cause with confidence is to have a look.
Are you sure that there's no access to the tub's plumbing from the back side or below? If not, consider installing an access panel at an appropriate location in your wall or ceiling.
This is a DIY site, so if you're not up to that task as a starting point, you probably need a professional.